# To fill 12 vacancies there are 25 candidates of which five are from scheduled caste. If three of the vacancies are reserved for scheduled caste candidates while the rest are open to all, the number of ways in which the selection can be made is

(a) ${}^5{C_3} \times {}^{22}{C_9}$

(b) ${}^{22}{C_9} - {}^5{C_3}$

(c) ${}^{22}{C_3} + {}^5{C_3}$

(d) None of these

Answer

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Hint: We know that selection of r things out of n things is given by ${}^n{C_r}$ . In the above question, first of all we have to select 3 vacancies reserved for scheduled caste for 5 scheduled caste candidates. After allotting three vacancies, the remaining vacancies will be filled by 22 candidates. Then we can find the number of ways to select the remaining 9 out of 22 candidates. Finally multiplying these two results will get us our answer.

We have been given that there are 12 vacancies and 25 candidates out of which five are from scheduled caste. If three of the vacancies are reserved for scheduled caste candidates then we have to find the number of ways in which the selection can be made.

Since, order of selection is not necessary. So, we will take formula of combination of r things out of n things given by,

${}^n{C_r}$

Now, there are 3 vacancies reserved for scheduled caste for 5 scheduled caste candidates.

The number of ways to select 3 candidates out of 5 $ = {}^5{C_3}$

Now, after allotting 3 vacancies to scheduled caste, we have \[\left( {{\rm{12}} - {\rm{3}}} \right) = {\rm{9}}\] more vacancies to fill by remaining candidates.

Remaining candidates $ = 25 - 3$

$ = 22$

So, we can write the number of ways to select 9 candidates out of 22 $ = {}^{22}{C_3}$

Now, we can multiply the number of ways of selecting 3 candidates from scheduled caste and the number of ways of selecting the remaining 9 candidates to get the total number of ways for selecting candidates as below,

Total number of ways for the selection $ = {}^{22}{C_3} \times {}^5{C_3}$

Therefore, the correct option is ‘a’.

Note: The mistake we generally do while finding the total number of ways in which the selection can be made, we add the number of ways to select 3 candidates and the number of ways to select 9 out of the remaining candidates. Thus, we get the incorrect answer. Just remember that in this type of question, if the condition is compulsory, then, we have to multiply the different ways of selection. Here, filling the vacancies is compulsory.

__Complete step-by-step solution -__We have been given that there are 12 vacancies and 25 candidates out of which five are from scheduled caste. If three of the vacancies are reserved for scheduled caste candidates then we have to find the number of ways in which the selection can be made.

Since, order of selection is not necessary. So, we will take formula of combination of r things out of n things given by,

${}^n{C_r}$

Now, there are 3 vacancies reserved for scheduled caste for 5 scheduled caste candidates.

The number of ways to select 3 candidates out of 5 $ = {}^5{C_3}$

Now, after allotting 3 vacancies to scheduled caste, we have \[\left( {{\rm{12}} - {\rm{3}}} \right) = {\rm{9}}\] more vacancies to fill by remaining candidates.

Remaining candidates $ = 25 - 3$

$ = 22$

So, we can write the number of ways to select 9 candidates out of 22 $ = {}^{22}{C_3}$

Now, we can multiply the number of ways of selecting 3 candidates from scheduled caste and the number of ways of selecting the remaining 9 candidates to get the total number of ways for selecting candidates as below,

Total number of ways for the selection $ = {}^{22}{C_3} \times {}^5{C_3}$

Therefore, the correct option is ‘a’.

Note: The mistake we generally do while finding the total number of ways in which the selection can be made, we add the number of ways to select 3 candidates and the number of ways to select 9 out of the remaining candidates. Thus, we get the incorrect answer. Just remember that in this type of question, if the condition is compulsory, then, we have to multiply the different ways of selection. Here, filling the vacancies is compulsory.

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