Math CentralQuandaries & Queries


Question from Ravi, a student:

In how many ways can the letters of the word STRANGE be arranged to that the vowels occupy only the old places.

Hi Ravi,

You have the arrangement

      A     E

and you are to fill the empty places with the letters STRNG, in any order.

Start by placing one of the letters in the leftmost position. There are 5 choices for the letter you select. REgardless of the letter you choose there are 4 letters remaining and hence you have 4 choices for the letter to place beside the first letter you chose. Thus there are $5 \times 4$ choices for the first two letters. Again, regardless of which two letters you have already used there are 3 letters remaining and hence to fill the first 3 positions you have $5 \times 4 \times 3$ choices. Repeat this procedure for the remaining two positions to arrive at the total number of choices.

If you are also allowed to have the arrangement

      E     A

then you will need to double the total number of possible arrangements.




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