Track / Stage 4 / 120 of 340 #380 of 1964

Problem 380

AMC 12 late, AIME early

Combinatorics Difficulty 4.8 Multiple choice

4. Let $A=\{a, b, c, d\}, B=\{1,2,3\}$ . If the mapping $f: A \rightarrow B$ , such that
$f(a)+f(b)+f(c)+f(d)=8,$

then the number of such mappings is:

Pick one

Official solution

4.C.

Consider the number of integer solutions to $x_{1}+x_{2}+x_{3}+x_{4}=8\left(1 \leqslant x_{i} \leqslant 3, i=1,2,3\right.$ , $4)$ , which is equivalent to the number of integer solutions to $y_{1}+y_{2}+y_{3}+y_{4}=4\left(0 \leqslant y_{i} \leqslant 2\right.$ , $i=1,2,3,4)$ . It equals $\mathrm{C}_{7}^{3}-4 \mathrm{C}_{4}^{3}=19$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.