Permutation and CombinationMedium
Question
The number of seven-digit numbers, that can be formed by using the digits $1,2,3,5$ and 7 such that each digit is used at least once, is ;
Options
A.15400
B.17800
C.16800
D.29400
Solution
Sol. [Key Concept: Split into 'exactly one digit repeated' and 'exactly two repeated' cases and use multinomial arrangements.]
Exactly one digit repeated $\rightarrow \ ^{5}C_{1} \times \frac{7!}{3!} = 4200$
Exactly two digits repeated $\rightarrow \ ^{5}C_{2} \times \frac{7!}{2!2!} = 12600$ Total numbers $= 16800$
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