Permutations & Combinations
Distribution of Objects
Grade 11
Question:
<p>Number of ways in which 12 different things can be distributed in 5 sets of 2, 2, 2, 3, 3 things is</p>
<p>(a) \(\frac{12!}{(3!)^2(2!)^3}\)</p>
<p>(b) \(\frac{12! \cdot 5!}{(3!)^2(2!)^3}\)</p>
<p>(c) \(\frac{12!}{(3!)^3(2!)^4}\)</p>
<p>(d) \(\frac{12! \cdot 5!}{(3!)^2(2!)^4}\)</p>
Step-by-Step Solution
Key Concept: When distributing into indistinct groups of the same size, divide by the factorial of the number of such groups.
<p>We have 3 groups of size 2 and 2 groups of size 3. Since groups of the same size are indistinguishable, we divide by $(3!)^2$ for the two groups of 3 and $(2!)^3$ for the three groups of 2. The answer is $\frac{12!}{(3!)^2(2!)^3}$.</p>
Correct Answer: A