The number of seven-digit numbers, that can be formed by using the digits 1, 2, 3, 4, 5 and 7 such that each digit is used at least once, is: [2026]

Question

The number of seven-digit numbers, that can be formed by using the digits 1, 2, 3, 4, 5 and 7 such that each digit is used at least once, is: [2026]

Smart Achievers · Accepted Answer

(3)

Case 1: One digit appears 3 times and other four digits appear once each.

Number of such arrangements

$= {}^{5}C_{1} \times \frac{7!}{3!}$

$= \frac{5 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1} = 4200$

Case 2: Two digits appear 2 times each and other three digits appear once each.

Number of such arrangements

$= {}^{5}C_{2} \times \frac{7!}{2! 2!}$

$= \frac{5 \times 4}{2} \times \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2}{2 \times 2} = 12600$

Total number of arrangements

= 4200 + 12600 = 12800

Solution

Q.
The number of seven-digit numbers, that can be formed by using the digits 1, 2, 3, 4, 5 and 7 such that each digit is used at least once, is: [2026]

1 15400

2 17800

3 16800

4 29400

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