Sum

Find the number of different arrangements of letters in the word MAHARASHTRA. How many of these arrangements have letters M and T never together?

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#### Solution

Total number of letters in the word MAHARASHTRA = 11

The letter ‘A’ is repeated ‘4’ times.

The letter ‘H’ is repeated twice.

The letter ‘R’ is repeated twice.

∴ Number of arrangements = `(11!)/(4!2!2!)`

Other than M and T. there are 9 letters in which A repeats 4 times, H repeats twice, R repeats twice

The number of arrangements of the letter = `(9!)/(4!2!2!)`

These 9 letters create 10 gaps in which M and T are to be arranged

The number of the arrangement of M and T = ^{10}P_{2}

∴ Total number arrangement having M and T never together = `(9!xx^10"P"_2)/(4!2!2!)`

Concept: Permutations When All Objects Are Not Distinct

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