Permutations And Combinations

Question
CBSEENMA11014163

In how many of the distinct permutations of the latter in MISSISSIPPI do the four I’s not come together? 

Solution

The letter of the word MISSISSIPPI are:
                      M → 1
                      I  → 4
                      S → 4
                      P → 2
              Total   → 11
rightwards double arrow n = 11, p = 4, q = 4, r = 2
∴ The number of permutations =  fraction numerator straight n factorial over denominator straight p factorial space straight q factorial space straight r factorial end fraction space equals space fraction numerator 11 factorial over denominator 4 factorial space 4 factorial space 2 factorial end fraction equals fraction numerator 11 cross times 10 cross times 9 cross times 8 cross times 7 cross times 6 cross times 5 cross times 4 cross times 3 cross times 2 cross times 1 over denominator 4 cross times 3 cross times 2 cross times 1 cross times 4 cross times 3 cross times 2 cross times 1 cross times 2 cross times 1 end fraction
                                         = 11 x 10 x 9 x 7 x 5 = 34650.
  Tie the 4 1's together.
  Number of permutations = space space fraction numerator straight P presuperscript 4 subscript 4 over denominator 4 factorial end fraction
(∵ 4 I's are similar)
                                  = fraction numerator 4 factorial over denominator 4 factorial end fraction equals 1                                                                          ...(i)
Mix the bundle of I's with IM + 4S + 2P to give a total of (1 + 4 + 2) + 1 = 8
∴ The number of permutations = fraction numerator 8 factorial over denominator 4 factorial space 2 factorial end fraction space equals space fraction numerator 8 cross times 7 cross times 6 cross times 5 cross times 4 cross times 3 cross times 2 cross times 1 over denominator 4 cross times 3 cross times 2 cross times 1 cross times 2 cross times 1 end fraction equals 840.
Hence, the number of permutations, when the four I's are together = 840 x 1 = 840
The number of permutations when the four I's are not all together
(Total permutations) - (permutations when I's are together) = 34650 - 840 = 33810.
                 

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