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  4. How many arrangements can be made out of the letters of the word MOTHER taken four at a time so that each arrangement contains the letter ´M´?

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Q. How many arrangements can be made out of the letters of the word " MOTHER" taken four at a time so that each arrangement contains the letter ´M´?

Permutations and Combinations

Solution:

There are six letters in MOTHER, all different, i.e. arrangement can be made out of the letters of the word MOTHER taken four at a time with M present in every arrangement.
So, rest 3 letters can be arrangement from 5 letters So, total number of ways $= 4 × ^5P_3$
$=4\times\frac{5!}{\left(5-3\right)!}=\frac{4\times5\times4\times3\times2}{2}=240$