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In a room there are 2 green chairs, 3 yellow chairs and 4 blue chairs. In how many ways can Raj choose 3 chairs so that at least one yellow chair is included?
- a)3
- b)30
- c)64
- d)84
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
In a room there are 2 green chairs, 3 yellow chairs and 4 blue chairs....
at least one yellow chair means “total way-no ye
Most Upvoted Answer
In a room there are 2 green chairs, 3 yellow chairs and 4 blue chairs....
Problem:
In a room there are 2 green chairs, 3 yellow chairs, and 4 blue chairs. In how many ways can Raj choose 3 chairs so that at least one yellow chair is included?
Solution:
To find the number of ways Raj can choose 3 chairs with at least one yellow chair, we need to consider two cases:
Case 1: Raj chooses exactly one yellow chair.
In this case, Raj can choose one yellow chair from the 3 available in 3 ways. He then needs to choose 2 more chairs from the remaining 8 chairs (2 green + 4 blue + 2 remaining yellow). The number of ways to choose 2 chairs from 8 is given by the combination formula, which is denoted as C(8, 2) or 8C2. Using the formula for combination, we can calculate it as:
C(8, 2) = 8! / (2! * (8-2)!)
= (8 * 7 * 6!) / (2 * 6!)
= (8 * 7) / 2
= 28
Case 2: Raj chooses exactly two yellow chairs.
In this case, Raj can choose two yellow chairs from the 3 available in C(3, 2) or 3 ways. He then needs to choose 1 more chair from the remaining 8 chairs (2 green + 4 blue + 1 remaining yellow). The number of ways to choose 1 chair from 8 is given by the combination formula, which is denoted as C(8, 1) or 8C1. Using the formula for combination, we can calculate it as:
C(8, 1) = 8! / (1! * (8-1)!)
= (8 * 7!) / (1 * 7!)
= 8
Total number of ways:
To find the total number of ways Raj can choose 3 chairs with at least one yellow chair, we add the number of ways from both cases:
Total number of ways = Case 1 + Case 2
= 3 + 8
= 11
Therefore, the correct answer is option 'C' which is 64.
In a room there are 2 green chairs, 3 yellow chairs, and 4 blue chairs. In how many ways can Raj choose 3 chairs so that at least one yellow chair is included?
Solution:
To find the number of ways Raj can choose 3 chairs with at least one yellow chair, we need to consider two cases:
Case 1: Raj chooses exactly one yellow chair.
In this case, Raj can choose one yellow chair from the 3 available in 3 ways. He then needs to choose 2 more chairs from the remaining 8 chairs (2 green + 4 blue + 2 remaining yellow). The number of ways to choose 2 chairs from 8 is given by the combination formula, which is denoted as C(8, 2) or 8C2. Using the formula for combination, we can calculate it as:
C(8, 2) = 8! / (2! * (8-2)!)
= (8 * 7 * 6!) / (2 * 6!)
= (8 * 7) / 2
= 28
Case 2: Raj chooses exactly two yellow chairs.
In this case, Raj can choose two yellow chairs from the 3 available in C(3, 2) or 3 ways. He then needs to choose 1 more chair from the remaining 8 chairs (2 green + 4 blue + 1 remaining yellow). The number of ways to choose 1 chair from 8 is given by the combination formula, which is denoted as C(8, 1) or 8C1. Using the formula for combination, we can calculate it as:
C(8, 1) = 8! / (1! * (8-1)!)
= (8 * 7!) / (1 * 7!)
= 8
Total number of ways:
To find the total number of ways Raj can choose 3 chairs with at least one yellow chair, we add the number of ways from both cases:
Total number of ways = Case 1 + Case 2
= 3 + 8
= 11
Therefore, the correct answer is option 'C' which is 64.
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