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JEE MainMathematicsModeratePemutation and combination

How many five letter words are to be formed out of the letters of
the word INFINITESIMAL

Show answer and solution
13 letters (4I, 2n, 7 Diff) 9 type
All different  9C55!=15120\ ^{9}C_{5}5! = 15120
2 alike 3 different  2C1. 8C35!2!=6720\ ^{2}C_{1}.\ ^{8}C_{3}\frac{5!}{2!} = 6720
3 alike 2 alike 1 different  2C2. 7C15!2!2!=210\ ^{2}C_{2.}\ ^{7}C_{1}\frac{5!}{2!2!} = 210
3 alike 2 different  1C1. 8C25!3!=5603\ alike\ 2\ different\ \ ^{1}C_{1}.\ ^{8}C_{2}\frac{5!}{3!} = 560
3 alike 2 alike  1C1. 1C15!3!2!=10\ \ ^{1}C_{1}.\ ^{1}C_{1}\frac{5!}{3!2!} = 10
4 alike 1 different  1C1. 8C15!4!=40\ \ ^{1}C_{1}.\ ^{8}C_{1}\frac{5!}{4!} = 40
Total =

Source: Grass-G / M-11 Mathematics Paper-2 1Nov.docx

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