## Unexam #2

Given in class on January 5, 2006.

Q: How many integral solutions are there to the equation
x_1 + x_2 + x_3 + x_4 + x_5 = 20 with x_i>0?

A: 3876

Q: How many integral solutions are there to the equation
x_1 + 2 x_2 + 3 x_3 + 4 x_4 + 5 x_5 = 20 with x_i=>0?

A: 192

Q: How many pairs of positive integers (a,b) are there with
a*b = 1555200 = 2^8*3^5*5^2
?

A: 162

You should answer all three of these questions.  I am again looking for a solution which explains these answers completely.  I would like you to be direct and terse and I do not want to see irrelevant details in your solution.  The first unexam I gave was relatively simple compared to this one but the idea is exactly the same and I will be using very similar criteria when I grade it.

These questions are stated more in terms of a number theory problem, but there are equivalent combinatorial questions for the first two that you might recognize that we have 'done before' (or at least questions similar).

Q1: How many ways are there of distributing 20 identical pieces of candy to 5 different children so that each child gets at least one piece of candy?

Q2: How many ways are there of making change for \$20 using bills with denominations of size \$1, \$2, \$3, \$4, \$5?

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