The number of 4-letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is

Question

The number of 4-letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is ________.

Smart Achievers · Accepted Answer

(2454)

The word EXAMINATION has $2 N, 2 A, 2 I, E, X, M, T, O$

Case I: If all letters are different, then number of words

$= {}^{8}P_{4} = \frac{8!}{4!} = 8 \times 7 \times 6 \times 5 = 1680$

Case II: If 2 letters are same and 2 are different, then number of words

$= {}^{3}C_{1} \cdot {}^{7}C_{2} \cdot \frac{4!}{2!} = 3 \times 21 \times 12 = 756$

Case III: If 2 letters are same of one kind and 2 letters are same of different kind, then number of words

$= {}^{3}C_{2} \cdot \frac{4!}{2! 2!} = 3 \times 6 = 18$

$∴$ Total number of words $= 1680 + 756 + 18 = 2454$

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