Stars and bars confusion, they have a slightly different answer to me. I have a problem:

A family has six children, A,B,C,D,E,F. The ages of these six children are added together to make an integer less than $32$ where none of the children are less than one year old.

What is the chances of guessing the ages of all of the children.
I thought stars and bars, and this is equivalent to choosing $7$ children for $$x_1+\dots+x_7=31$$
So we get (1 in)${31-1 \choose 7-1}$
But they said that $x_1+\dots+\dots+x_7=32$ is correct. I feel like that would be the case if the original question said $x_1+\dots+x_6\leq32$, but it said $x_1+\dots+x_6\lt32$
So apparently that makes it (1 in)${32-1 \choose 7-1}$
Which is right?(Probably them I know, but I want to know why mine is wrong please)
 A: When you set the sum at $31$, you’re allowing the extra variable $x_7$ to be $0$. However, the formula that you’re using is for the number of solutions in strictly positive integers, so you want to make sure that $x_7$ will have to be at least $1$. You do this by insisting that the total be $32$ rather than $31$.
A: We want to solve $x_1+\cdots+x_7=32$, where all the $x_i$ are $\ge 1$. For $x_7\ge 1$, which means $x_1+\cdots +x_6\lt 32$.
If you use $31$, then since $x_7\ge 1$, the ages of the real children would be forced to add up to less than $31$.
Remark: This answers the Stars and Bars question. However, that question has little connection with the motivating question about the chances of guessing the ages of the children. For one thing the ages are a monotone sequence of integers, with equalities relatively unlikely. 
A: given $$\sum_i^6x_i<32\implies \sum_i^7x_i=31(\because x_7\ge0,x_{(i,0\le i\le6)}\ge1)$$
let $$x_7'=x_7+1,x_7'\ge1$$
so $$\sum_i^6x_i+x_7'=32$$
now all $x_i\ge1$, now you can solve it maybe?
