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Find the number of all possible arrangement of the letters of the word “MATHEMATICS” taken four from at a time.

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The word MATHEMATICS consist of 11 letters: 

(M,M), (A,A), (T,T), H,E,I,C,S 

Case 1: In this case 2 similar and 2 similar letters are selected, number of arrangements 

= 3C2 × \(\frac{4!}{2!\space2!}=756\) 

Case 2: In this case all 4 letters selected are different, number of arrangements = 8C4 × 4! 

= 1680 

Therefore, total number of arrangement = 18 + 756 + 1680 

= 2454

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