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Solution


Q.

The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is __________ .     [2023]

Ans.

(413)

Case 1: 1 1 1 1 1 4 3

Number of ways = 7!5!=42                    [Since, 1 repeats five times]

Case 2: 1 1 1 1 3 3 2

Number of ways = 7!4!2!=105             [Since, 1 repeats 4 times and 3 repeats 2 times]

Case 3: 2 2 2 3 1 1 1

Number of ways =  7!3!3!=140            [Since, 2 repeats 3 times and 1 repeats 3 times]

Case 4: 2 2 2 2 2 1 1

Number of ways = 7!5!2!=21                [Since, 2 repeats 5 times and 1 repeats 2 times]

Case 5: 1 1 1 1 4 2 2

Number of ways = 7!4!2!=105              [Since, 1 repeats 4 times and 2 repeats 2 times]

∴   Total number of ways=42+105+140+21+105=413