Question

A box contains two white, three black and four red balls, In how many ways can three be drawn from the box if at least one black ball is to be included in the draw?

• A. If balls of the same colour are identical then $6$
• B. If balls of the same colour are identical then $8$
• C. If balls of the same colour are different then $74$
• D. If balls of the same colour are different then $64$

Answer 1

Answer: (A), (D)

The correct options are
A If balls of the same colour are identical then $6$
D If balls of the same colour are different then $64$

If balls of same colour are identical

3 Balls

1 Black $\to$ (3 possibilities ${}_{RR}^{WW}RW$ )

2 Black $\to$ (2 possibilities W (or) R)

3 Black $\to$ only 1

So Total 6

If balls of same color are different

then

1 Black (2 white) ${\to }^3{C}_1{\times }^2{C}_2$

1 Black (2 red) ${\to }^3{C}_1{\times }^4{C}_2$

1 Black (1 Red 1 white) ${\to }^3{C}_1{\times }^4{C}_1{\times }^2{C}_1$

2 Black (Red white) ${\to }^3{C}_2{\times }^4{C}_1$

${\to }^3{C}_2{\times }^2{C}_1$

3 Black ${\to }^3{C}_3$

Total $=3+18+24+12+6+1$

$=64$