Q6Single correct4 Marks3 Sep 2023

A box contains 2 red pens, 3 black pens and 4 blue pens. In how many ways can 3 pens be drawn from the box, if at least one black pen is to be included?

A

$9$

B

$64$

C

$84$

D

$20 We want to draw 3 pens such that at least one is black. - Total number of ways to draw any 3 pens from 9 is \binom{9}{3} = (9 \cdot 8 \cdot 7) / (3 \cdot 2 \cdot 1) = 84. - Total number of ways to draw 3 pens with NO black pens (meaning we only choose from the 2 red + 4 blue = 6 non-black pens) is \binom{6}{3} = (6 \cdot 5 \cdot 4) / (3 \cdot 2 \cdot 1) = 20. - Number of ways to draw 3 pens with at least one black pen is: Total ways - Ways with no black pens = 84 - 20 = 64.$