how may four letter words can be formed using the letters of the word FAILURE so that F is included in each word?

Options

A. `""^(6)C_(4)xx4!`
B. `""^(6)C_(3)xx4!`
C. `""^(6)C_(4)`
D. `""^(6)C_(4)xx3!`

Answer

B

Explanation

There are 7 letters in the words 'FAILURE'. To include F in every 4 letter words, we first select four letters from the 7 letters of the word 'FAILURE' such that F is included in every selection. This can be done by selecting three letters from the remainig 6 letters i.s., A, I, L, U, R, E, in `""^(6)C_(3)` ways. Now, there are 4 letters in each of `""^(6)C_(3)` selections.
Consider one of these `""^(6)C_(3)` selections. this selection contains 4 letters which can be arranged in 4! ways. Thus each of `""^(6)C_(3)` selections provides 4! words.
Hence, the total number of words `""^(6)C_(3)xx4!`