Question:
A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?
A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?
Updated On: Sep 3, 2024
- 64
- 129
- 84
- None of these
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The Correct Option is A
Solution and Explanation
At least one black ball can be drawn in the following ways
(i) one black and two other colour balls
$={ }^{3} C_{1} \times{ }^{6} C_{2}=3 \times 15=45$
(ii) two black and one other colour balls
$={ }^{3} C_{2} \times{ }^{6} C_{1}=3 \times 6=18$
(iii) All the three are black $={ }^{3} C_{3} \times{ }^{6} C_{0}=1$
$\therefore$ Re no. of ways $=45+18+1=64$
(i) one black and two other colour balls
$={ }^{3} C_{1} \times{ }^{6} C_{2}=3 \times 15=45$
(ii) two black and one other colour balls
$={ }^{3} C_{2} \times{ }^{6} C_{1}=3 \times 6=18$
(iii) All the three are black $={ }^{3} C_{3} \times{ }^{6} C_{0}=1$
$\therefore$ Re no. of ways $=45+18+1=64$
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Concepts Used:
Combinations
The method of forming subsets by selecting data from a larger set in a way that the selection order does not matter is called the combination.
- It means the combination of about ‘n’ things taken ‘k’ at a time without any repetition.
- The combination is used for a group of data where the order of data does not matter.
- For example, Imagine you go to a restaurant and order some soup.
- Five toppings can complement the soup, namely:
- croutons,
- orange zest,
- grated cheese,
- chopped herbs,
- fried noodles.
But you are only allowed to pick three.
- There can be several ways in which you can enhance your soup with savory.
- The selection of three toppings (subset) from the five toppings (larger set) is called a combination.
Use of Combinations:
It is used for a group of data (where the order of data doesn’t matter).