alleryThe number of positive integral solutions of $x_1+x_2+\cdots+x_r=n$ is
A${}^{n+r-1}C_{r-1}$
B${}^{n-1}C_{r-1}$
C${}^{n}C_r$
D${}^{n+r}C_r$
Answer & Solution
Correct answer: B. ${}^{n-1}C_{r-1}$
For positive integral solutions, set $y_i=x_i-1\ge 0$.
Then $y_1+y_2+\cdots+y_r=n-r$.
The number of non-negative solutions is ${}^{(n-r)+r-1}C_{r-1}={}^{n-1}C_{r-1}$.
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