Let =(4 !) ! (4 !)^ 3 ! and =(5 !) ! (5 !)^ 4 ! . Then : - Vidyadip

MCQ

Let $\alpha=\frac{(4 !) !}{(4 !)^{3 !}}$ and $\beta=\frac{(5 !) !}{(5 !)^{4 !}}$. Then :

A
$\alpha \in \mathrm{N}$ and $\beta \notin \mathrm{N}$
B
$\alpha \notin \mathrm{N}$ and $\beta \in \mathrm{N}$
✓
$\alpha \in \mathrm{N}$ and $\beta \in \mathrm{N}$
D
$\alpha \notin \mathrm{N}$ and $\beta \notin \mathrm{N}$

✓

Answer

Correct option: C.

$\alpha \in \mathrm{N}$ and $\beta \in \mathrm{N}$

c
$\alpha=\frac{(4 !) !}{(4 !)^{3 !}}, \beta=\frac{(5 !) !}{(5 !)^{4 !}} $

$ \alpha=\frac{(24) !}{(4 !)^6}, \beta=\frac{(120) !}{(5 !)^{24}}$

Let $24$ distinct objects are divided into $6$ groups of $4$ objects in each group.

No. of ways of formation of group $=\frac{24 !}{(4 !)^6 \cdot 6 !} \in \mathrm{N}$

Similarly,

Let $120$ distinct objects are divided into $24$ groups of $5$ objects in each group.

No. of ways of formation of groups

$=\frac{(120) !}{(5 !)^{24} \cdot 24 !} \in N$

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