1991 AJHSME Problems/Problem 11

Problem

There are several sets of three different numbers whose sum is $15$ which can be chosen from $\{ 1,2,3,4,5,6,7,8,9 \}$ . How many of these sets contain a $5$ ?

$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7$

Solution

Let the three-element set be $\{ a,b,c \}$ and suppose that $a=5$ .

We need $b+c=10$ and $b\neq c$ . This gives us four solutions, so there are $4$ sets with a $5$ also with the desired properties $\rightarrow \boxed{\text{B}}$ .

1991 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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1991 AJHSME Problems/Problem 11

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