Question

A committee of 4 persons is to be formed from 2 ladies, 2 old men and 4 young men such that it includes at least one lady. At least one old man and at most 2 young men. Then the total number of ways in which committee can be formed is:
 $
  {\text{A}}{\text{. 40}} \\
  {\text{B}}{\text{. 41 }} \\
  {\text{C}}{\text{. 16 }} \\
  {\text{D}}{\text{. 32}} \\
  $

Answer 1

Hint: To answer this question first we have to think about possibilities and count all the possibilities one by one then using permutation and combination we will calculate and reach the required option.

Complete step-by-step answer:
We have 2 ladies, 2 old men and 4 young men and we have to select 4 according to the conditions given.
Now possibilities are
Young man ladies old man
0 2 2
1 1 2
1 2 1
2 1 1
Here all possibilities are taken according to the condition that we have to select at most 2 young men, at least 1 lady and at least 1 old man.
So we have
(0 ym, 2L, 2 om) + ( 1 ym, 1 L, 2om) + ( 1 ym, 2L, 1om) + ( 2ym, 1L, 1 om)
Here ym means young man , L means ladies , and om means old man.
Now using concept of permutation combination we can write
 $ ^4{C_0}{.^2}{C_2}{.^2}{C_2}{ + ^4}{C_1}{.^2}{C_1}{.^2}{C_2}{ + ^4}{C_1}{.^2}{C_2}{.^2}{C_1}{ + ^4}{C_2}{.^2}{C_1}{.^2}{C_1} $
Here $ ^4{C_0} $ is used we have to select 0 young man from 4 young man and $ ^2{C_2} $ because we have to select 2 ladies from 2 ladies and $ ^2{C_2} $ for selection of 2 old man from given 2 old man. Same concept is applied for further selections.
Here $ ^4{C_0} $ = 1 you can find out using formula $ ^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!\left( {r!} \right)}} $ or we may directly write 1 because we can understand that if we have to select no one from given 4 then there is one and only one way of doing this.
 $
  1 \times 1 \times 1 + 4 \times 2 \times 1 + 4 \times 1 \times 2 + 6 \times 2 \times 2 \\
   = 41 \\
  $
Hence option B is the correct option.

Note: Whenever we get this type of question the key concept of solving is we have to have knowledge of the permutation combination like if we have n objects and we have to select r from them then the total number of possibilities will be $ ^n{C_r} $ . And the best thing which should be cared for is possibilities we have to count every case.