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6.Permutation and Combination
hard
The number of ways five alphabets can be chosen from the alphabets of the word $MATHEMATICS$, where the chosen alphabets are not necessarily distinct, is equal to :
A
$175$
B
$181$
C
$177$
D
$179$
(JEE MAIN-2024)
Solution
$AA$, $MM$, $TT$, $H$, $I$, $C$, $S$, $E$
($1$) All distinct
${ }^8 \mathrm{C}_5 \rightarrow 56$
($2$) $2$ same, $3$ different
${ }^3 \mathrm{C}_1 \times{ }^7 \mathrm{C}_3 \rightarrow 105$
($3$) 2 same $I^{\text {st }}$ kind, 2 same $2^{\text {nd }}$ kind, 1 different
${ }^3 \mathrm{C}_2 \times{ }^6 \mathrm{C}_1 \rightarrow 18$
Total $\rightarrow 179$
Standard 11
Mathematics
