Question

A company has 5 men and 6 women. What are the number of ways of selecting a group of eight persons?
A.165
B.185
C.205
D.225

Answer 1

 Hint:Calculate the total number of persons i.e. n and then assume r as eight as per the given condition. Then use the formula ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$ to get the number of ways of selecting a group of eight students.

Complete step by step answer:
As we have to find the number of ways of selecting a group of eight persons we will write the given data first,
Number of men in the company = 5
Number of women in the company = 6
Therefore total workers in the company = n = 5 + 6 = 11
As we have to find the number of ways for eight persons therefore it will become our ‘r’ in the combination,
Therefore, r = 8
As we have to find the number of ways of selecting a group of eight persons therefore we have to find eight number of combinations from total eleven persons, therefore we will get,
Number of ways of selecting a group of eight persons = ${}^{11}{{C}_{8}}$ ………………………………… (1)
Now to proceed further in the solution we should know the formula given below,
Formula:
${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$
By using the formula given above we will write the equation (1) as follows,
Therefore, number of ways of selecting a group of eight $=\dfrac{11!}{8!\left( 11-8 \right)!}$
Now by doing subtraction in the denominator we will get,
Therefore, number of ways of selecting a group of eight $=\dfrac{11!}{8!\times 3!}$
Therefore, number of ways of selecting a group of eight $=\dfrac{11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{\left( 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1 \right)\times \left( 3\times 2\times 1 \right)}$
Therefore, number of ways of selecting a group of eight $=\dfrac{11\times 10\times 9}{3\times 2}$
Therefore, number of ways of selecting a group of eight $=\dfrac{11\times 5\times 9}{3}$
Therefore, number of ways of selecting a group of eight $=11\times 5\times 3$
Therefore, number of ways of selecting a group of eight $=165$
Therefore, the number of ways of selecting a group of eight persons is equal to 165
Therefore the correct answer in option (a).

Note: While solving these type of problems there might be confusion between the formula of \[{}^{n}{{C}_{r}}\] and \[{}^{n}{{P}_{r}}\] therefore do remember that the formula of \[{}^{n}{{C}_{r}}\] is \[\dfrac{n!}{r!\left( n-r \right)!}\] so that you can avoid silly mistakes in the exam.