Find the number of combinations in the letters of the word STATISTICS taken $4$ at a time.

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Solution

In the word $" S T A T I S T I C S "$
$S = 3$
$T = 3$
$A = 1$
$I = 2$
$C = 1$
$C - I$ (Three letters are same)
$2_{C_{1}} \times 4_{C_{1}} \frac{4!}{3!} = 32$
$C - I I$ (Two letter are same and other two letter are same)
$3_{C_{1}} \times \frac{4!}{2! 2!} = 18$
$C - I I I$ (Two letter are same & other two letter are different)
$3_{C_{1}} \times 4_{C_{2}} \times \frac{4!}{2!} = 216$
$C - I V 5_{C_{4}} \times 4! = 120$
$∴$ Total number of words $= 386$

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Find the number of combinations in the letters of the word STATISTICS taken 4 at a time.

In the word "STATISTICS"S=3T=3A=1I=2C=1C−I (Three letters are same)2C1×4C14!3!=32C−II (Two letter are same and other two letter are same)3C1×4!2!2!=18C−III (Two letter are same & other two letter are different)3C1×4C2×4!2!=216C−IV5C4×4!=120∴ Total number of words =386

Find the number of combinations in the letters of the word STATISTICS taken $4$ at a time.