# 1985 AHSME Problems/Problem 11

## Problem

How many distinguishable rearrangements of the letters in $CONTEST$ have both the vowels first? (For instance, $OETCNST$ is one such arrangement but $OTETSNC$ is not.) $\mathrm{(A)\ } 60 \qquad \mathrm{(B) \ }120 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 720 \qquad \mathrm{(E) \ }2520$

## Solution

We can separate each rearrangement into two parts: the vowels and the consonants. There are $2$ possibilities for the first value and $1$ for the remaining one, for a total of $2\cdot1=2$ possible orderings of the vowels. There are similarly a total of $5!=120$ possible orderings of the consonants. However, since both T's are indistinguishable, we must divide this total by $2!=2$. Thus, the actual number of total orderings of consonants is $120 \div 2=60$. Thus In total, there are $2\cdot60=120$ possible rearrangements, $\boxed{\text{(B)}}$.

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