How many $5$ letter arrangements can be made from the word “numeracy” if it MUST include the letter “$y$”?

Question

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?

Answer 1

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.

Answer 2

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

Answer 3

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 $\binom 74$.