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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?

- 60
- 40
- 64
- 82

64

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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 (^{4}C_{2}) = 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 (^{3}C_{2}) = 3 ways |
1 (^{2}C_{1}) = 2 ways |
0 | 3 × 2 = 6 ways |

2 (^{3}C_{2}) = 3 ways |
0 | 1 (^{4}C_{1}) = 4 ways |
3 × 4 = 12 ways |

Total |
18 ways |

When 3 Black Shirts are selected.

BLACK |
WHITE |
RED |
TOTAL |

3 (^{3}C_{3}) = 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? " **

Choice C is the correct answer.

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