How many permutations are possible with 8 letters ‘a’, ‘b’, ‘c’, ‘g’, ‘h’, ‘p’, ‘q’ and 'm'?

Question

CONSTRAINT= 3rd and 7th letter of the permutation is always 'm’ and ‘h’ Please clarify I am confused with 6! ways

Answer 1

Recall that to perform the task of laying out the letters, you do not need to perform the steps in any specific order. Approach with rule of product and begin by picking the values for the more restrictive steps first:

• Pick what letter appears in the third position: This must be the 'm' and so there is only one option

• Pick what letter appears in the seventh position: This must be the 'h' and so there is only one option

• Pick what letter appears in the first position: This must be some remaining letter different than 'm' or 'h', so there are six options

• Pick what letter appears in the second position: This must be some remaining letter different than 'm', 'h', or whatever was picked in the previous step, so there are five options

• Pick what letter appears in the fourth position: This must be some remaining letter different than 'm', 'h', or whatever was picked in either of the previous two steps, so there are four options remaining

• $\vdots$

Multiplying the number of options gives the total: $1\times 1\times 6\times 5\times 4\times 3\times 2\times 1 = 6!$