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 [2024]

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Solution

Q.
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 [2024]

1 175

2 181

3 179

4 177

Ans.
(3)

We have, MMAATTHEICS

Case-I : All distinct $= {}^{8}{C_{5}} = 56$

Case-II : 2 identical + 3 distinct $= {}^{3}{C_{1} \times {}^{7}{C_{3} = 105}}$

Case-III : 2 identical + 2 identical + 1 distinct $= {}^{3}{C_{2} \times {}^{6}{C_{1}} = 18}$

Hence, total number of ways = 56 + 105 + 18 = 179

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