How many $4$ digits numbers can due formed by using the digits $1, 2, 3, 4, 5, 6, 7$ if at least one digit is repeated.

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Solution

Total no. of digits $= 7$
No. of ways $4$ digit numbers can be formed with repetition $= 7 \times 7 \times 7 \times 7 = 2401$
No. of ways $4$ can be formed with repetition $= 7 \times 6 \times 5 \times 4 = 840$
Therefore,
No. of ways $4$ digit numbers can be formed if atleast one digit is repeated $= 2401 - 840 = 1561$

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