Permutations and Combinations – Arrangements – JEE Main 2 Apr 2025 Shift 1

Solution:

We are looking for the number of sequences of length 10 using digits {0, 1, 2}.

Given:
Number of 1s = 5

Number of 2s = 3

The remaining terms must be 0s.

Number of 0s = $10 – (5 + 3) = 2$

The number of such sequences is the number of arrangements of these 10 items (where 5 are alike of one kind, 3 are alike of another, and 2 are alike of a third kind): 11111 222 00
$$ \text{Total sequences} = \frac{10!}{5! \cdot 3! \cdot 2!} $$

Expand the factorials to simplify:
$$ = \frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5!}{5! \cdot (3 \cdot 2 \cdot 1) \cdot (2 \cdot 1)} $$

$$ = \frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6}{6 \cdot 2} $$

$$ = 10 \cdot 9 \cdot 4 \cdot 7 = 2520 $$

Ans. (3)