- Home
- Standard 11
- Mathematics
6.Permutation and Combination
hard
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
