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The following comes from questions comes from a recent combinatorics paper I attended :

1.27 people are to travel by a bus which can carry 12 inside and 15 outside. In how many ways can the party be distributed between inside and outside if 5 people refuse to go outside and 6 will not go inside?

The solution given C(16,7), I have no clue how they got it ?!

2.The number of functions f from the set A = {0, 1, 2} into the set B = {1, 2, 3, 4, 5, 6, 7} such that $f(i) \le f(j) $ for $i \lt j $ and $i,j$ belongs to A is

The solution given is C(8,3). I didn't really understood this one.

3.The number of ordered pairs $(m, n) m, n $ is in {1 , 2, … , 100} such that $7^m + 7^n$ is divisible by 5 is

The solution given is 2500, but how ?

4.The coefficient of $x^{20}$in the expansion of $(1 + 3x + 3x^2 + x^3)^{20}$, is ?

How to solve this one elegantly ?

5.An eight digit number divisible by 9 is to be formed by using 8 digits out of the digits 0, 1, …, 9 without replacement. The number of ways in which this can be done is:

Now this one seems impossible for me to solve in 1 mint,or is it ? Given soln is 36(7!)

  • 8
    Ha, I was wondering for a second what "1.27 people" meant... ;)2010-10-24
  • 2
    Nice question, but _please_ split such questions up into pieces to be answered individually next time. Also, the title is quite unenlightening.2013-02-02
  • 1
    @HansLundmark, made my day. +12013-02-02

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