Permutations & Combinations
Multinomial Division — Natural Number Check
nta_pyq_2024_jan
Grade 11
Question:
Let $\alpha=\dfrac{(4!)!}{(4!)^{3!}}$ and $\beta=\dfrac{(5!)!}{(5!)^{4!}}$. Then:
$\alpha\in\mathbb{N}$ and $\beta\notin\mathbb{N}$
$\alpha\notin\mathbb{N}$ and $\beta\in\mathbb{N}$
$\alpha\in\mathbb{N}$ and $\beta\in\mathbb{N}$
$\alpha\notin\mathbb{N}$ and $\beta\notin\mathbb{N}$
Step-by-Step Solution
Key Concept: $\alpha=\frac{24!}{(4!)^6}$: number of ways to divide 24 distinct objects into 6 ordered groups of 4 — a multinomial coefficient, hence $\in\mathbb{N}$. $\beta=\frac{120!}{(5!)^{24}}$: similarly a multinomial coefficient, hence $\in\mathbb{N}$.
$\alpha=\frac{24!}{(4!)^6}\in\mathbb{N}$, $\beta=\frac{120!}{(5!)^{24}}\in\mathbb{N}$.
Correct Answer: 3