Number of ways in which four letters of word ‘MATHEMATICS ’ can be arranged is given by

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6.Permutation and Combination

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The number of ways in which four letters of the word $‘MATHEMATICS$’ can be arranged is given by

A

$136$

B

$192$

C

$1680$

D

$2454$

Solution

(d) Word $‘MATHEMATICS$’ has $2M, 2T, 2A, H, E, I, C, S.$ Therefore $4$ letters can be chosen in the following ways.

Case $I : 2$ alike of one kind and $2 $ alike of second kind

$i.e.$,$^3{C_2} \Rightarrow $ No. of words ${ = ^3}{C_2}\frac{{4\;!}}{{2\;!\;2\;!}} = 18$
Case $ II$ : $2$ alike of one kind and $2$ different

i.e.,$^3{C_1}{ \times ^7}{C_2} \Rightarrow $No.of words ${ = ^3}{C_1}{ \times ^7}{C_2} \times \frac{{4\;!}}{{2\;!}} = 756$

Case $III$ : All are different

$i.e.$,$^8{C_4} \Rightarrow $No. of words ${ = ^8}{C_4} \times 4\;! = 1680$.

Hence total number of words are $2454$.

Standard 11

Mathematics

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