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)". be as specific and detailed as possible.
how many solutions does the equation have, x1 + x2 + x3 = 10, where x1, x2, and x3 are non-negative integers?
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 - 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

Step-by-step explanation:

Given is an equation in three variables

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

Thus possible solutions are 36                              

no

step-by-step explanation:

easy its decreasing

step-by-step explanation:

because its top point is at the to of the graph on the left side (atleast from the veiw i see it) now if that top point wast in the lower part of the graph it would be increasing hope this have a nice day