# 6-letter strings from the word APPLES where L comes before E? [closed]

Consider the word APPLES. How many 6-letter strings are possible such that the L appears to the left of (but not necessarily next to) the E?

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## closed as off-topic by Grigory M, mrf, Vedran Šego, tetori, Austin MohrDec 26 '13 at 2:29

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There are $$\left(\frac{6!}{2!}\right)$$ to make six-letter words. Divide by two to find the number of words in which L is to the left of E. So, the answer is $$\frac{\left(\frac{6!}{2!}\right)}{2} = 180$$

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Small corection, It should be $\left( \frac{6!}{2!} \right)$ and not $\left(\frac{6}{2}\right)$ –  user44197 Dec 25 '13 at 23:29

In total, there are $\frac{6!}{2!}$ arrangements of these six letters (we divide by $2!$ since there are two letters P). Half of these have the L to the right of the E, and half of them have the L to the left of the E.

Answer: $\frac{6!}{2\cdot 2!}=180$.

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How do you know that half have the L to the right of the E and half of them have the L to the left of the E? –  user112829 Dec 25 '13 at 21:47
There is a trivial bijection between "arrangements with L to the left of E" and "arrangements with L to the right of E", so they must be equally many. –  Mårten W Dec 25 '13 at 21:49
There are $\binom{6}{2}$ ways to choose the two places for the L and E, and then $\frac{4!}{2!}$ ways to arrange A,P,P,S in the remaining 4 places; so there are $\binom{6}{2}\cdot\frac{4!}{2!}=15\cdot12=180$ ways in total.