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What I’ve done so far- I know that $y$ will occupy one spot of $5$ letters. I also know that I have to multiply by $5$ to get the final answer. However, I don’t know how to proceed to find the value of y when it is any slot. Can anybody help me answer this one?

Righter
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Johnathan B. Abbott
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  • Choose the position for the y. Choose which of the remaining letters goes in the first open position, which of the remaining letters goes in the first remaining open position, and so forth until you run out of open positions. If you are familiar with combinations, you could simply choose which four other letters will be used in the arrangement, then arrange the five letters, which include y, in the five positions. – N. F. Taussig Apr 08 '21 at 23:15

3 Answers3

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First, forget the y in numeracy and find the number of ways to pick $4$ letters from numerac. There are $7 \choose 4$ ways, or $35$. Each way can be rearranged $4! =24$ times, so the total number of arrangements is $35*24 =840$. Now, look at an example $4$-letter arrangement NUME. You can put a Y in the following spaces in _ N _ U _ M _ E _ to make a $5$-letter word with Y. There are $5$ spaces, so you multiply $840*5$ to get $4200$ ways.

Some Guy
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  • Since y must be included, a modification of this argument is to select four additional letters from the set ${\text{n, u, m, e, r, a c}}$, which can be done in $\binom{7}{4}$ ways, then permute the five selected letters, which can be done in $5!$ ways, which yields $\binom{7}{4}5! = 4200$ ways. – N. F. Taussig Apr 09 '21 at 09:11
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Slightly more straightforwardly than the already clear chosen answer, your five letters will be Y along with four chosen from NUMERAC, so $\binom 74$ choices of which letters to use. These chosen letters can then be arranged in $5!$ ways:$$\binom 74 5! = 35\cdot 120$$

One reason this question is straightforward is that there are no repeated letters to deal with (see this problem for an example of dealing with that issue). The restriction to force the selection of Y just reduces the letter choice from $\binom 85$ to $\binom 74$.

Joffan
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If y is in the first slot, decide how many letters can be chosen for the second, then the third and so on.

John Hunter
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