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Q.
How many four letter words can be formed using the letters of the word FAILURE, so that F is included in each word?
JamiaJamia 2010
Solution:
There are 7 letters in the word FAILURE. To include F in every 4 letters word, 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 remaining six letters. ie, $ A,\text{ }I,\text{ }L,\text{ }U,\text{ }R,\text{ }E $ in $ ^{6}{{C}_{3}} $ ways. Now there are 4 letters in each of $ ^{6}{{C}_{3}} $ selections. Also every selection contains letters which can be arranged in 4! ways. Hence, the required number of four letter words $ {{=}^{6}}{{C}_{3}}\times 4!=\frac{6.5.4}{3.2.1}.24 $ $ =20\times 24=480 $