(D) The word $MATHEMAGICA$ consists of $11$ letters.The frequency of each letter is: $M: 2, A: 3, T: 1, H: 1, E: 1, G: 1, I: 1, C: 1$.The total number

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(D) The word $MATHEMAGICA$ consists of $11$ letters.The frequency of each letter is: $M: 2, A: 3, T: 1, H: 1, E: 1, G: 1, I: 1, C: 1$.The total number of permutations is given by the formula $\frac{n!}{n_1! n_2! \dots n_k!}$.Substituting the values: $\frac{11!}{2! 3!} = \frac{11 \times 10 \times 9 \times 8 \times 7!}{2 \times 1 \times 6} = \frac{7920}{12} \times 7! = 660 \times 7!$.Thus,the correct option is $D$.

Class 11 Mathematics Permutation and Combination Definition of permutation, Number of permutations with or without repetition, Conditional permutations

How many permutations can be made by using all the letters of the word $MATHEMAGICA$?

A
$11!$
B
$(420)(8!)$
C
$\frac{11!}{2!2!}$
D
$(660)7!$

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Permutation and Combination

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Definition of permutation, Number of permutations with or without repetition, Conditional permutations

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How many permutations can be made by using all the letters of the word $MATHEMAGICA$?

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There are $10$ students in a classroom,including $3$ girls $A, B,$ and $C$. In how many ways can they be arranged in a row such that no two of the three girls are together?

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