JEE Main 2024 · 27 January · evening · Single correct · +4 / −1
Let α=((4 !) !)/(4 !)^(3 !) and β=((5 !) !)/(5 !)^(4 !). Then :
- (a) α ∈ N and β ∈ N
- (b) α ∈ N and β ∉ N
- (c) α ∉ N and β ∈ N
- (d) α ∉ N and β ∉ N
Answer and solution
Answer (a) α ∈ N and β ∈ N
Solution
Maths sir · 4 steps
Do not expand these monsters. Read each one as a multinomial coefficient and integrality comes free.
- The key fact. Split mn distinct objects into n labelled groups of m each. The number of ways is a count, so it is a natural number.
(mn)!/(m!)ⁿ = C(mn, m, m, …, m) ∈ N
- Rewrite alpha. Here 4! = 24 = 4 × 6 and 3! = 6. So α = (4 × 6)!/(4!)⁶, which is the case m = 4, n = 6: 24 objects in 6 groups of 4. Hence α ∈ N.
- Rewrite beta. Here 5! = 120 = 5 × 24 and 4! = 24. So β = (5 × 24)!/(5!)²⁴, which is the case m = 5, n = 24: 120 objects in 24 groups of 5. Hence β ∈ N.
- So. Both α and β are natural numbers, which is option (a).
The idea to keep. Whenever you see (mn)!/(m!)ⁿ, it counts ways to split mn objects into n groups of m, so it is always an integer. In general (n!)!/(n!)^((n-1)!) fits this pattern.
Answer from the NTA final answer key. Solution by PrepWiser.