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:
Solution
<p>$$\begin{aligned}
& 2 M \\
& 2 A \\
& 2 T \\
& H, E, I, C, S
\end{aligned}$$</p>
<p>Case-I</p>
<p>2 Alike 2 Alike 1 Diff</p>
<p>${ }^3 C_2 \times{ }^6 C_1=18$</p>
<p>Case-II</p>
<p>2 Alike + 3 Diff</p>
<p>${ }^3 C_1 \times{ }^7 C_3=105$</p>
<p>Case-III</p>
<p>All different</p>
<p>${ }^8 C_5=56$</p>
<p>Total ways $=179$</p>
About this question
Subject: Mathematics · Chapter: Permutations and Combinations · Topic: Fundamental Counting Principle
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