Question

m dustinct animals of a circus have to be placed in m cages, one in each cage. There are n small cages and p small animals (n < p < m). The large anumals are so large that they do not fit in small cage, However, small animals can be put in any cage, The number of ways of putting the animals into cages is:
1. \(\{^{m - n}P_p\}\{^{m - n}P_{m - p}\}\)
2. \(^{m - n}C_p\)
3. \(\{^{m - n}C_p\}\{^{m - n}C_{m - p}\}\)
4. \(^{m - n}P_p\)

Answer 1

Correct Answer - Option 1 : \(\{^{m - n}P_p\}\{^{m - n}P_{m - p}\}\)

Concept:

Number of ways to choose r things out of n things is given by ⁿC_r

Number of ways to arrange r things out of n things is given by ⁿP_r

 

Calculation:

Number of large cages = m - n

Number of big animals = p

Here, ways to arrange big animals(p) in m, n cages = ^m-nP_p 

Number of remaining cages = n - p

Now, remaining animals can be arranged as, ^n-pP_m-p

So, total number of ways = \(\{^{m - n}P_p\}\{^{m - n}P_{m - p}\}\)

Hence, option (1) is correct.