A) 136
B) 192
C) 1680
D) 2454
Correct Answer: D
Solution :
Word ?MATHEMATICS? has 2M, 2T, 2A, H, E, I, C, S. Therefore 4 letters can be chosen in the following ways. Case I : 2 alike of one kind and 2 alike of second kind \[i.e.\],\[^{3}{{C}_{2}}\Rightarrow \]No. of words \[{{=}^{3}}{{C}_{2}}\frac{4\ !}{2\ !\ 2\ !}=18\] Case II : 2 alike of one kind and 2 different i.e.,\[^{3}{{C}_{1}}{{\times }^{7}}{{C}_{2}}\Rightarrow \]No.of words \[{{=}^{3}}{{C}_{1}}{{\times }^{7}}{{C}_{2}}\times \frac{4\ !}{2\ !}=756\] Case III : All are different \[i.e.\],\[^{8}{{C}_{4}}\Rightarrow \]No. of words \[{{=}^{8}}{{C}_{4}}\times 4\ !=1680\]. Hence total number of words are 2454.
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