Question

From $4$ officers and $8$ jawans, in how many ways can $6$ be chosen such that it includes exactly one officer.

Answer 1

Hint: Calculate the number of ways to choose $1$ officers from $4$ officers. Calculate the number of ways to choose $6-1=5$ jawans from $8$ jawans. Multiply both the values to calculate the total number of ways to choose $6$ people.

Complete step-by-step answer:
We have a group of $4$ officers and $8$ jawans. We have to choose $6$ people such that it includes exactly one officer. We have to find the number of ways to do so.
We will calculate the number of ways to choose one officer from $4$ officers. Then we will calculate the number of ways to choose remaining people from $8$ jawans. We will then multiply both the values to calculate the total number of ways to choose $6$ people.
We know there are ${}^{n}{{C}_{r}}$ ways to choose $r$ people from a group of $n$ people. We know that ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$.
Substituting $n=4,r=1$, we have ${}^{4}{{C}_{1}}=\dfrac{4!}{1!\left( 3 \right)!}=\dfrac{4\times 3!}{3!}=4$ ways to choose one officer from $4$ officers.
As we have to choose total $6$ people and we have already chosen one officer, the number of jawans to be chosen $=6-1=5$. So, we will now choose $5$ jawans from $8$ jawans.
Substituting $n=8,r=5$, we have ${}^{8}{{C}_{5}}=\dfrac{8!}{5!\left( 3 \right)!}=\dfrac{8\times 7\times 6\times 5!}{5!\times 3!}=\dfrac{8\times 7\times 6}{3\times 2}=56$ ways to choose $5$ jawans from $8$ jawans.
To calculate the total number of ways to choose $6$ people according to the given data, we will multiply both the values of choosing one officer from $4$ officers and $5$ jawans from $8$ jawans.
Thus, the total number of ways to choose $6$ people according to the given data $=4\times 56=224$.
Hence, we can choose $6$ people such that it includes exactly one officer is $224$.

Note: In this question, we are basically calculating all the possible combinations to choose people. One must keep in mind that we are not considering the arrangement of chosen people on this question. If we count the arrangement of people, we will get an incorrect answer.