The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen…
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 :
179
177
181
175
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 $\rightarrow 179$