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There are 6 cages to keep 6 lions in a circus. One cage is so small that any of the four big lions cannot be kept in them. Find the number of ways in which 6 lions can be kept in 6 cages.

Last updated date: 23rd Jul 2024
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Hint: For solving this question you should know about permutation and combinations. In this problem we will use these for finding the number of ways to keep the lions. We will use the combination in this to find our required number of ways. And we solve it to get the required solution.

Complete step by step answer:
According to our question it is said that there are 6 cages to keep 6 lions in a circus. One cage is so small that any of the four big lions cannot be kept in them and we have to find the number of ways in which 6 lions can be kept in 6 cages. As we know that permutations and combinations are generally used for solving these types of questions. The permutation is used for lists and in permutation, the order matters and combinations are used for groups and in that the order does not matter. Permutation relates to the act of arranging all the members of a set into some sequence or order. And combinations is a way of selecting items from a collection such that (unlike permutations) the order of selection does not matter.
According to the given question,
Number of ways to fill a small cage = $^{2}{{C}_{1}}$
The rest cages filled by the 5 remaining lions = $5!$
Therefore, the total ways is,
\begin{align} & ={}^{2}{{C}_{1}}\times 5! \\ & =2\times 5! \\ & =2\times 5\times 4\times 3\times 2\times 1 \\ & =2\times 120 \\ & =240 \\ \end{align}
So, the total ways are 240.

Note: While solving these types of questions you should be careful about selecting permutation or combination. You have to ensure that which has to be used in the given problem, because if the selected method is wrong, then the answer will also be wrong.