Permutation and Combination

No. of ways of selecting r objects out of n objects = nCr
No. of ways of arranging n objects = n!
No. of ways of arranging n objects with a identical & b identical = n!
a! b!
No. of ways of arranging r objects out of n objects
= n(n–1)(n–2) ... (n – (r – 1)) = nPr = nCr r!
No. of ways of arranging n objects in a circle = (n – 1)!

Sample questions:
1. Choosing People:
Que: A team of 4 is to be chosen from a group consisting of Anne and Bob and 4 other people. In how
many ways can this be done if
(i) there are no restrictions?
(ii) Anne must be in the team?
(iii) Anne and Bob must both be in the team?
(iv) at most one of Anne and Bob in the team?
(v) Anne or Bob or both are in the team?
Ans:
(i) No. of ways of choosing 4 people = 6C4 = 15
(ii) No. of ways of choosing the other 3 people = 5C3 = 10
(iii) No. of ways of choosing the other 2 people = 4C2 = 6
(iv) Total no. of ways – no. of ways with Anne & Bob both in the team = 15 – 6 = 9
(v) No. of ways with Anne in the team + no. of ways with Bob in the team – no. of ways with
both in the team = 10 + 10 – 6 = 14

2. Choosing from Different Types of People (e.g. Boys & Girls)
Que: A team of 3 is to be chosen from a group of 3 boys and 4 girls. How many ways can this be done if
(i) there are no restrictions?
(ii) there must be exactly 1 boy?
(iii) there must be at least 1 boy?
(iv) there must be at least 1 boy and at least 1 girl?

Ans: (i) No. of ways of choosing 4 people = 7C3 = 35
(ii) No. of ways of choosing 1 boy and 2 girls = 3C1 4C2 = 18
(iii) No. of ways = Total no. of ways – no. of ways with no boys = 35 – 4C3 = 35 – 4 = 31
Note: It is wrong to say no. of ways = 3C1 6C2 = 45
(iv) No. of ways = Total no. of ways – no. of ways with no boys – no. of ways with no girls
= 35 – 4C3 – 3C3 = 35 – 4 – 1 = 30
Note: It is wrong to say no. of ways = 3C1 4C1 5C1 = 60

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Divisibility rules

These rules let you test if one number is divisible by another, without having to do too much calculation!
A number is divisible by,
2 If the last digit is even, the number is divisible by 2.
3 If the sum of the digits is divisible by 3, the number is also.
4 If the last two digits form a number divisible by 4, the number is also.
5 If the last digit is a 5 or a 0, the number is divisible by 5.
6 If the number is divisible by both 3 and 2, it is also divisible by 6.
7 Take the last digit, double it, and subtract it from the rest of the number; if the answer is divisible by 7 (including 0), then the number is also.
8 If the last three digits form a number divisible by 8,
then so is the whole number.
9 If the sum of the digits is divisible by 9, the number is also.
10 If the number ends in 0, it is divisible by 10.
11 Alternately add and subtract the digits from left to right. (You can think of the first digit as being 'added' to zero.) If the result (including 0) is divisible by 11, the number is also. Example: to see whether 365167484 is divisible by 11, start by subtracting: [0+]3-6+5-1+6-7+4-8+4 = 0; therefore 365167484 is divisible by 11.
12 If the number is divisible by both 3 and 4, it is also divisible by 12.
13 Delete the last digit from the number, then subtract 9 times the deleted digit from the remaining number. If what is left is divisible by 13,
then so is the original number.

Another Rule For 11,
Subtract the first digit from a number made by the other digits.
If that number is divisible by 11 then the original number is, too.
Can repeat this if needed,

Example: 286,
28 − 6 is 22, which is divisible by 11, so 286 is divisible by 11
Example: 14641
1464 − 1 is 1463
146 − 3 is 143
14 − 3 is 11, which is divisible by 11, so 14641 is divisible by 11

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Math: The First 1000 Primes

The 1st is 2, 100th is 541 and the 1,000th is 7919

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759 1777 1783 1787 1789 1801 1811 1823 1831 1847 1861 1867 1871 1873 1877 1879 1889 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987 1993 1997 1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089 2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293 2297 2309 2311 2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411 2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543 2549 2551 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663 2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851 2857 2861 2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 2963 2969 2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089 3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221 3229 3251 3253 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331 3343 3347 3359 3361 3371 3373 3389 3391 3407 3413 3433 3449 3457 3461 3463 3467 3469 3491 3499 3511 3517 3527 3529 3533 3539 3541 3547 3557 3559 3571 3581 3583 3593 3607 3613 3617 3623 3631 3637 3643 3659 3671 3673 3677 3691 3697 3701 3709 3719 3727 3733 3739 3761 3767 3769 3779 3793 3797 3803 3821 3823 3833 3847 3851 3853 3863 3877 3881 3889 3907 3911 3917 3919 3923 3929 3931 3943 3947 3967 3989 4001 4003 4007 4013 4019 4021 4027 4049 4051 4057 4073 4079 4091 4093 4099 4111 4127 4129 4133 4139 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231 4241 4243 4253 4259 4261 4271 4273 4283 4289 4297 4327 4337 4339 4349 4357 4363 4373 4391 4397 4409 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493 4507 4513 4517 4519 4523 4547 4549 4561 4567 4583 4591 4597 4603 4621 4637 4639 4643 4649 4651 4657 4663 4673 4679 4691 4703 4721 4723 4729 4733 4751 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831 4861 4871 4877 4889 4903 4909 4919 4931 4933 4937 4943 4951 4957 4967 4969 4973 4987 4993 4999 5003 5009 5011 5021 5023 5039 5051 5059 5077 5081 5087 5099 5101 5107 5113 5119 5147 5153 5167 5171 5179 5189 5197 5209 5227 5231 5233 5237 5261 5273 5279 5281 5297 5303 5309 5323 5333 5347 5351 5381 5387 5393 5399 5407 5413 5417 5419 5431 5437 5441 5443 5449 5471 5477 5479 5483 5501 5503 5507 5519 5521 5527 5531 5557 5563 5569 5573 5581 5591 5623 5639 5641 5647 5651 5653 5657 5659 5669 5683 5689 5693 5701 5711 5717 5737 5741 5743 5749 5779 5783 5791 5801 5807 5813 5821 5827 5839 5843 5849 5851 5857 5861 5867 5869 5879 5881 5897 5903 5923 5927 5939 5953 5981 5987 6007 6011 6029 6037 6043 6047 6053 6067 6073 6079 6089 6091 6101 6113 6121 6131 6133 6143 6151 6163 6173 6197 6199 6203 6211 6217 6221 6229 6247 6257 6263 6269 6271 6277 6287 6299 6301 6311 6317 6323 6329 6337 6343 6353 6359 6361 6367 6373 6379 6389 6397 6421 6427 6449 6451 6469 6473 6481 6491 6521 6529 6547 6551 6553 6563 6569 6571 6577 6581 6599 6607 6619 6637 6653 6659 6661 6673 6679 6689 6691 6701 6703 6709 6719 6733 6737 6761 6763 6779 6781 6791 6793 6803 6823 6827 6829 6833 6841 6857 6863 6869 6871 6883 6899 6907 6911 6917 6947 6949 6959 6961 6967 6971 6977 6983 6991 6997 7001 7013 7019 7027 7039 7043 7057 7069 7079 7103 7109 7121 7127 7129 7151 7159 7177 7187 7193 7207 7211 7213 7219 7229 7237 7243 7247 7253 7283 7297 7307 7309 7321 7331 7333 7349 7351 7369 7393 7411 7417 7433 7451 7457 7459 7477 7481 7487 7489 7499 7507 7517 7523 7529 7537 7541 7547 7549 7559 7561 7573 7577 7583 7589 7591 7603 7607 7621 7639 7643 7649 7669 7673 7681 7687 7691 7699 7703 7717 7723 7727 7741 7753 7757 7759 7789 7793 7817 7823 7829 7841 7853 7867 7873 7877 7879 7883 7901 7907 7919 end.

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Math: Perfect Square

How to guess if the number is a perfect square:

1. A perfect square will always have a unit digit of 1, 4, 5, 6, 9 or 0.

2. A perfect square will always have a 10′s digit which is even, Except when the unit digit of a perfect square is 6, in which case the 10’s digit is always odd.

3. If the unit digit of a perfect square is any other number, then the 10’s digit of a perfect square is always even.

4. A perfect square ending in 5, will always have 2 as the 10’s digit.

5. A perfect square ending in 0, will always have 0 as the 10’s digit ( even number of ending 0’s) .

 A number cannot be an exact or perfect square if:

- it ends in 2, 3,7 or 8

- it terminates in an odd number of zeros

- its last digit is 6 but its penultimate (tens) digit is even

- its last digit is not 6 but its penultimate (tens) digit is odd

- its last digit is 5 but its penultimate (tens) digit is other than 2

- its last 2 digits are not divisible by 4 if it is even number

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Math formulas: Algebra (PDF)

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Math Formulas: Partnership

Partnership:

When two or more than two persons run a business jointly, they are called partners and the deal is known as partnership.

Ratio of Divisions of Gains:

When investments of all the partners are for the same time, the gain or loss is distributed among the partners in the ratio of their investments.

Suppose A and B invest Rs. x and Rs. y respectively for a year in a business, then at the end of the year:

(A's share of profit) : (B's share of profit) = x : y

When investments are for different time periods, then equivalent capitals are calculated for a unit of time by taking (capital x number of units of time). Now gain or loss is divided in the ratio of these capitals.

Suppose A invests Rs. x for p months and B invests Rs. y forq months then,

(A's share of profit) : (B's share of profit)= xp : yq

Working and Sleeping Partners:

A partner who manages the the business is known as a working partner and the one who simply invests the money is a sleeping partner.

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Math Formulas: Allegation or Mixture

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Math Formulas: H.C.F and L.C.M

Factors and Multiples:

If number a divided another number b exactly, we say that a is afactor of b.

In this case, b is called a multiple of a.

Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.): The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly.

There are two methods of finding the H.C.F. of a given set of numbers:

Factorization Method: Express the each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F.

Division Method: Suppose we have to find the H.C.F. of two given numbers, divide the larger by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is required H.C.F.

Finding the H.C.F. of more than two numbers: Suppose we have to find the H.C.F. of three numbers, then, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given number.

Similarly, the H.C.F. of more than three numbers may be obtained.

Least Common Multiple (L.C.M.):

The least number which is exactly divisible by each one of the given numbers is called their L.C.M.

There are two methods of finding the L.C.M. of a given set of numbers:

Factorization Method: Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors.

Division Method (short-cut): Arrange the given numbers in a rwo in any order. Divide by a number which divided exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers.

Product of two numbers = Product of their H.C.F. and L.C.M.

Co-primes: Two numbers are said to be co-primes if their H.C.F. is 1

H.C.F. and L.C.M. of Fractions:

1. H.C.F. = H.C.F. of Numerators/ L.C.M. of Denominators

2. L.C.M. = L.C.M. of Numerators/ H.C.F. of Denominators

H.C.F. and L.C.M. of Decimal Fractions:

In a given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.

Comparison of Fractions:

Find the L.C.M. of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with L.C.M as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.

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Math Formulas: Problems on Speed, Time and Distance

1. km/hr to m/s conversion:
    a km/hr = a x (5/18) m/s
2. m/s to km/hr conversion:
    a m/s = a x (18/5)km/hr

 Formulas for finding Speed, Time and Distance

1. Time taken by a train of length l meters to pass a pole or standing man or a signal post is equal to the time taken by the train to cover lmeters.
2. Time taken by a train of length l meters to pass a stationery object of length b meters is the time taken by the train to cover (l + b) meters.
3. Suppose two trains or two objects bodies are moving in the same direction at u m/s and v m/s, where u > v, then their relative speed is = (u - v) m/s.
4. Suppose two trains or two objects bodies are moving in opposite directions at u m/s and v m/s, then their relative speed is = (u + v) m/s.
5. If two trains of length a meters and b meters are moving in opposite directions at u m/s and v m/s, then: 
   The time taken by the trains to cross each other = (a + b)/(u + v) sec
6. If two trains of length a meters and b meters are moving in the same direction at u m/s and v m/s, then: 
   The time taken by the faster train to cross the slower train = (a +b)/(u - v) sec
7. If two trains (or bodies) start at the same time from points A and B towards each other and after crossing they take a and b sec in reaching B and A respectively, then:
   (A's speed) : (B's speed) = (√b : √a)

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Math Formulas: Profit and Loss

Cost Price:

The price, at which an article is purchased, is called its cost price, abbreviated as C.P.

Selling Price:

The price, at which an article is sold, is called its selling prices, abbreviated as S.P.

Profit or Gain:

If S.P. is greater than C.P., the seller is said to have a profit or gain.

Loss:

If S.P. is less than C.P., the seller is said to have incurred a loss.

IMPORTANT FORMULA

1. Gain = (S.P.) - (C.P.)
2. Loss = (C.P.) - (S.P.)
3. Loss or gain is always reckoned on C.P.
4. Gain Percentage: (Gain %)
        Gain % = (Gain x 100)/C.P.
5. Loss Percentage: (Loss %)
        Loss % = (Loss x 100)/C.P.
6. Selling Price: (S.P.)
        SP = ((100 + Gain %)/100) x C.P
7. Selling Price: (S.P.)
        SP = ((100 - Loss %)/100 x C.P
8. Cost Price: (C.P.)
        C.P. = ((100/(100 + Gain %) x S.P.
9. Cost Price: (C.P.)
        C.P. = ((100/(100 - Loss %) x S.P.
10. If an article is sold at a gain of say 35%, then S.P. = 135% of C.P.
11. If an article is sold at a loss of say, 35% then S.P. = 65% of C.P.

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Math Formulas: Time and Work

1. Work from Days:
   If A can do a piece of work in n days, then A's 1 day's work = 1/n
2. Days from Work:
   If A's 1 day's work = 1/n , then A can finish the work in n days.  
3. Ratio:
   If A is thrice as good a workman as B, then:
   Ratio of work done by A and B = 3 : 1.

   Ratio of times taken by A and B to finish a work = 1 : 3.

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গণিত সিরিজ/ধারা (সূত্রাবলী)

Formula:
১. 1+2+3+4+......+ n হলে এরূপ ধারার সমষ্টি = [n(n+1)/2]
২. প্রথম n পদের বর্গের সমষ্টি = [n(n+1)2n+1)/6]

৩. প্রথম n পদের ঘনের সমষ্টি = [n(n+1)/2]2
৪. পদ সংখ্যা = [(শেষ পদ – প্রথম পদ)/প্রতি পদে বৃদ্ধি] + ১
৫. সমষ্টি = [(১ম পদ + শেষ পদ)/২] x পদসংখ্যা
৬. n তম পদ = a + (n-1)d ‌এখানে, n = পদসংখ্যা, a = ১ম পদ, d = সাধারণ অন্তর
৭. n সংখ্যক পদের সমষ্টি = n/2[2a+(n-1)d]
৮. ১ম n সংখ্যক বিজোড় সংখ্যার সমষ্টি = n2
৯. ১ম n সংখ্যক জোড় সংখ্যার সমষ্টি = n(n+1)
১০. ১ম n সংখ্যক ক্রমিক ধারার গড় = (১ম পদ + শেষ পদ)/2

গুরুত্বপূর্ণ সমাধান:
* 1+2+3+4+…………+100 = ?
Solu: [n(n+1)/2] = [100(100+1)/2] = 5050
* 1^2+2^2+3^2+42+…………+50^2 = ?
Solu: [n(n+1)2n+1)/6] = [50(50+1)2x50+1)/6] = 42925
* 1^3+2^3+3^3+4^3+…………+10^3 = ?
Solu: [n(n+1)/2]2 = [10(10+1)/2]2 = 3025
* log2 + log4 + log8 + … ধারার 10টি পদের সমষ্টি কত?
Solu: log2 + log22 + log23 + … + log210
= (1+2+3+ … + 10)log2 = [10(10+1)/2] log2 = 55log2
* 5+10+15+…………+50 =?
Solu: পদসংখ্যা = [(শেষ পদ – প্রথম পদ)/প্রতি পদে বৃদ্ধি] + ১ = [(50 – 5)/5] + 1 = 10
সুতরাং, সমষ্টি = [(5 + 50)/2] x 10 = 275
* পরপর ১০টি সংখ্যা দে3য়া দেয়া আছে, ১ম ৫টির যোগফল ৫৬০ হলে, শেষ ৫টির যোগফল কত?
উত্তর: ৫৬০ + ৫২ = ৫৮৫
* পরপর ৬টি সংখ্যা দেয়া আছে, শেষ ৩টির যোগফল ৩৬ হলে, প্রথম ৩টির যোগফল কত?
উত্তর: ৩৬-৩২ = ২৭

Bonus:
ল.সা.গু এবং গ.সা.গু
* দুটি সংখ্যার গুণফল ১৫৩৬। সংখ্যা দুটির ল.সা.গু ৯৬ হলে, গ.সা.গু কত?
উত্তর: সংখ্যা দুটির গুণফল = ল.সা.গু x গু.সা.গু
বা, ১৫৩৬ = ৯৬ x গু.সা.গু
সুতরাং, গু.সা.গু = ১৬
* একটি ঘোড়ার গাড়ির সামনের চাকার পরিধি ৩ মি. পেছনের চাকার পরিধি ৪ মি.। গাড়িটি কত পথ গেলে সামনের চাকা পেছনের চাকার চেয়ে ১০০ বার বেশি ঘুরবে?
উত্তর: ৩ x ৪ x ১০০ = ১২০০ মিটার = ১.২ কি.মি.

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