JEE Main 2026 — Apr 2, Shift 1 — Mathematics solution

JEE Main2026Apr 2, Shift 1Mathematics

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 :

Solution

A seven-digit string using all five digits ${1, 2, 3, 5, 7}$ at least once must repeat two digits. There are two distribution patterns:

Pattern 1 (one digit appears thrice, the others once each: $3 + 1 + 1 + 1 + 1$ ):

Choose the digit to repeat: $(1 5) = 5$ ways. Arrange the multiset of seven items: $\frac{7 !}{3 !} = 840$ ways.

Subtotal $= 5 \times 840 = 4200$ .

Pattern 2 (two digits appear twice each, the others once each: $2 + 2 + 1 + 1 + 1$ ):

Choose the two repeating digits: $(2 5) = 10$ ways. Arrange: $\frac{7 !}{2 ! 2 !} = 1260$ ways.

Subtotal $= 10 \times 1260 = 12600$ .

Total $= 4200 + 12600 = 16800$ .

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