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Mathematics · Permutations and Combinations

JEE Main 2024 — 27 January, Shift 2 — Question 19

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

  1. Option A:

    α∈N\alpha \in \mathrm{N} and β∉N\beta \notin \mathrm{N}

  2. Option B:

    α∉N\alpha \notin \mathrm{N} and β∈N\beta \in \mathrm{N}

  3. Option C:

    α∈N\alpha \in N and β∈N\beta \in N

    Correct
  4. Option D:

    α∉N\alpha \notin \mathrm{N} and β∉N\beta \notin \mathrm{N}

Answer: C

Step-by-step solution

α=(4!)!(4!)3!,β=(5!)!(5!)4!\quad \alpha=\frac{(4!)!}{(4!)^{3!}}, \beta=\frac{(5!)!}{(5!)^{4!}}

α=(24)!(4!)6,β=(120)!(5!)24\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 =24!(4!)6.6!∈N=\frac{24!}{(4!)^{6} .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 =(120)!(5!)24⋅24!∈N=\frac{(120)!}{(5!)^{24} \cdot 24!} \in \mathrm{N}

Answer key and solution verified before publishing.

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Exam
JEE Main 2024
Paper
27 January, Shift 2
Subject
Mathematics
Chapter
Permutations and Combinations
Topic
Distribution of Objects into Groups
Let α=frac (4!)! (4!) 3! and β=frac (5!)! (5!) 4! . Then : | JEE Main 2024 PYQ with Solution · DhiX AI