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Question 12 : 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?

  1. 60
  2. 40
  3. 64
  4. 82

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Explanatory Answer

Two White Shirts - W1, W2
3 Black Shirts - B1, B2, B3
4 red Shirts - R1, R2, R3, R4

At least 1 black shirts - We cannot have 0 black shirts

When 1 Black Shirt is selected.

BLACK WHITE RED TOTAL
1 (3 ways) 0 (1 way) 2 (4C2) = 6 ways 3 × 1 × 6 = 18 ways
1 (3 ways) 2 (1 way) 0 3 × 1 = 3 ways
1 (3 ways) 1 (2 ways) 1(4 ways) 3 × 2 × 4 = 24 ways
Total 45 ways


When 2 Black Shirts are selected.
BLACK WHITE RED TOTAL
2 (3C2) = 3 ways 1 (2C1) = 2 ways 0 3 × 2 = 6 ways
2 (3C2) = 3 ways 0 1 (4C1) = 4 ways 3 × 4 = 12 ways
Total 18 ways


When 3 Black Shirts are selected.
BLACK WHITE RED TOTAL
3 (3C3) = 1 way 0 0 1 way
Total 1 way


So, Total number of ways
When 1 Black Shirt = 45 ways
When 2 Black Shirts = 18 ways
When 3 Black Shirts = 1 way

Total = 64 ways

The question is " 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? "

Hence, the answer is 64

Choice C is the correct answer.

 

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