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Q.
The number of ways in which four letters of the word MATHEMATICS can be arranged, is given by
Permutations and Combinations
Solution:
Available letters are MMAATTHEICS
There are $11$ letters in which $2 M's, 2 A's$ and $2 T’s$.
The following cases are possible: Case I : We have four letters having two pair of similar letters. This can be done in $^{3}C_{2}$ ways.
$\therefore $ Number of permutations of such four letters is $\frac{4!}{2! \,2!}$ Case II : One similar pair and $2$ different letters. This can be done in $ ^{3}C_{1} \times\,{}^{7}C_{2}$ ways and then number of possible permutations $\frac{4!}{ 2!}$ Case III : All different letters are taken. This can be done in $^{8}C_{4}$ ways and then number of permutations $= 4!$
$\therefore $ The required number is
$^{3}C_{2} \times\,\frac{4!}{2!\, 2!} +\,{}^{3}C_{1} \times\,{}^{7}C_{2} \times\, \frac{4!}{2!} +\,{}^{8}C_{4} \times\, 4! $
$= 2454$.