Question

I am studying for a discrete math final and I do not understand the provided solution. Specifically, how to rationalize "C(12,10) = C(12,2)". Please be as specific and detailed as possible. Thanks!
How many solutions does the equation have, x1 + x2 + x3 = 10, where x1, x2, and x3 are non-negative integers?

Answer: To count the number of solutions, we note that a solution corresponds to the ways of selecting 10 items fro m a set with three elements so that x1 items of type one, x2 items of type two, x3 items of type three. Hence, the number of solutions is equal to the number of 10 - Answer: combinations with repetitions allowed from a set with three elements. From Theorem 2, the equation has C(3 + 10−1, 10) = C(12,10) = C(12,2) = 66 solution

Answer 1

Answer:

Step-by-step explanation:

Given is an equation in three variables

$x_1+x_2+x_3 =10$ where the three variables are non negative integers.

Since sum =10, and each integer has minimum value 1, we find that each variable can take values as 1,2,...8 only

If first variable is 1, the other two can be 1,8  or 2,7 or... 8,1  ... 8 ways

If first variable is 2, ...   the other two can be 1,7  or 2,6 or..7,1 ... 7 ways

This will go up to first variable =8

Thus we have total number of solutions as

$8+7+6+5+...+1\\=36$

Thus possible solutions are 36