How many strings of length 9 can be formed using the letters HELLOJELLO?

Question

This is a two part question, the first part was how many strings of length 10 can be formed using the letter HELLOJELLO?

For this half I got $\frac{10!}{4!2!2!}.$ What is the difference in the question if we go from $10$ length to $9,$ how does the approach in solving it change?

Answer 1

One way is to break up into cases:

• a $H$ is left out
• an $E$ is left out
• an $L$ is left out
• an $O$ is left out
• a $J$ is left out

Solve for each individual case, and doing that is the same as how you would do it before, but with $9$ letters instead of $10$

Answer 2

The problem here is that, as the letters are repeated here, you can't just do C(10 9). You have to deal with all the cases differently.

You have E,O and L repeated twice, twice and four times respectively. You have to account for all these letters separately in their own cases.

• H is left out
• one E is left out
• one L is left out
• one O is left out
• J is left out

Now you can sum all of these cases' answers and that'll be your final answer.

Answer 3

Do by encasing each situation, like H left out or J left out or E left out or L left out or O left out.

H and J have similar cases and so does E and O.

When H or J left out = 2 * 9! / (4! * 2! * 2!) When E or O left out = 2 * 9! / (4! * 2!) When L is left out = 9! / (3! * 2! * 2!)

So adding them all up would finally give us 9! / 3! as our answer