Question

The number of ways so that $$3$$ men and $$2$$ women can sit in a bus such that the total number of sitting men and women on each side is $$3$$ ?
1. $$5!$$
2. $${}^6{C}_5\times 5!$$
3. $$6!\times {}^6{P}_5$$
4. $$5!+{}^6{C}_5$$

Answer 1

Hint: To solve this problem, first find out the total number of peoples in the bus and then recognize the total number of ways in which all peoples can sit after that find all the ways in which all seats can be filled and after that try to find number of ways in which all people can sit in a bus on each side.

Complete step-by-step solution:
Combination is a way of selecting items from a collection where the order of selection does not matter. The combination can be defined as “an arrangement of objects where the order of selected objects does not matter”.
For example: If we want to buy a milkshake and we are allowed to combine any two flavors from Apple, Banana and Cherry. So we are supposed to make a combination out of these possible flavors. We have total combinations as: Apple and banana, Apple and Cherry, Banana and Cherry. These are the only possible combinations from the given flavors and as we can see in this order of selected flavors do not matter.
As in smaller cases, it is possible for us to count the number of combinations, but when the cases have a large number of groups of elements, then we use a formula to determine all the possible selections. The formula is represented as:
$${}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{k}}=[\left(\mathbf{n}\right)(\mathbf{n}-\mathbf{1})(\mathbf{n}-\mathbf{2})\dots .(\mathbf{n}-\mathbf{k}+\mathbf{1})]/[(\mathbf{k}-\mathbf{1})(\mathbf{k}-\mathbf{2})\dots \dots .\left(\mathbf{1}\right)]$$
Now according to the given question:
Numbers of men in the bus are $$3$$
And numbers of women in the bus are $$2$$
So, total numbers of men and women in the bus equals to $$5$$
Therefore, total number of ways in which $$5$$ people will sit in bus is given as:
$$\Rightarrow 5!$$
Total numbers of seats to be filled in bus are $$6$$
So, total number of ways in which $$6$$ seats can be filled by $$5$$ people
$$\Rightarrow {}^6{C}_5$$
Therefore, total number of ways in which men and women can sit in a bus on each side is
$$\Rightarrow {}^6{C}_5\times 5!$$
Hence, the correct option from all above options is $$2$$.

Note: The difference between the permutation and combination is only of the order of the objects. In permutation the order of objects is very important, i.e. the objects must be in the proper order. In permutation there are only ordered elements whereas in the combination it can have unordered sets.