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# The number of ways in which four letters of the word $MATHEMATICS$ can be arranged is ?

$\begin{array}{1 1} 136 \\ 192 \\ 1680 \\ 2454 \end{array}$

Can you answer this question?

In the word $MATHEMATICS$ there are
$M$......2
$A$.......2
$T$.......2
H,E,I,C,S....5
Case (i) Four letters selected are :
Any two pairs of alphabets from M,A,T
The selection can be done in $^3C_2$ ways and then arranged in $\large\frac{4!}{2!.2!}$ ways.
Case (ii) Four letters selected are :
Two letters from any of the pairs of M,A,T and two from any of H,E,I,C,S and any 2 alphabets from M A,T other than the pair already selected.
The selection can be done in $^3C_1.^7C_2$ ways and
arranged in $\large\frac{4!}{2!}$
Case (iii) All the 4 letters are different.
The selection is done in $^8C_4$ ways and arranged in $4!$ ways.
$\therefore$ The required no. of arrangements
$=^3C_2.\large\frac{4!}{2!.2!}$$+^3C_1.^7C_2.\large\frac{4!}{2!}$$+^8C_4.4!$
$=3\times 6+3\times 21\times 12+70\times 24$
$=18+756+1680=2454$

answered Aug 15, 2013
edited Dec 23, 2013