A box contains 2 white shirts, 3 black shirts, and 4 red shirts. In how many ways can 3 shirts be drawn from the box, if at least one black shirt is to be included in the draw?

Name: A box contains 2 white shirts, 3 black shirts, and 4 red shirts. In how many ways can 3 shirts be drawn from the box, if at least one black shirt is to be included in the draw?
Uploaded: 2025-11-18T18:17:58.559986+05:30
Duration: 3 min 38 s
Description: A box contains 2 white shirts, 3 black shirts, and 4 red shirts. In how many ways can 3 shirts be drawn from the box, if at least one black shirt is to be included in the draw?

Solution

The possibilities of selecting 1 black shirt , 2 black shirts , 3 black shirts :

1. Ways to select all 3 black shirts only = 1
2. Ways to select 2 Black shirts : (B,B,W) & (B,B,R) = $$3_{C_2}$$($$2_{C_1}+\ 4_{C_1}$$) = 18
3. Ways to select 1 black shirt : (B,W,R) & (B,W,W) & (B,R,R) = (3)$$\left(2_{C_1}4_{C_1\ }+2_{C_2}+4_{C_2}\right)$$ = 45

Therefore, total ways = 1+18+45 = 64 ways.