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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?
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In a room there are 2 green chairs, 3 yellow chairs and 4 blue chairs....
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In a room there are 2 green chairs, 3 yellow chairs and 4 blue chairs....
To solve this problem, we can use the concept of combinations.
Step 1: Calculate the total number of ways to choose 3 chairs from all the available chairs
The total number of chairs in the room is 2 green chairs + 3 yellow chairs + 4 blue chairs = 9 chairs.
We need to choose 3 chairs from these 9 chairs, which can be done in C(9, 3) ways, where C(n, r) represents the number of combinations of choosing r items from a set of n items.
C(9, 3) = 9! / (3! * (9-3)!) = 9! / (3! * 6!) = (9 * 8 * 7) / (3 * 2 * 1) = 84 ways.
Step 2: Calculate the number of ways to choose 3 chairs without including any yellow chair
Since we want to find the number of ways to choose at least one yellow chair, we need to subtract the number of ways to choose 3 chairs without including any yellow chair from the total number of ways calculated in step 1.
To choose 3 chairs without including any yellow chair, we can choose from the green and blue chairs only.
The number of ways to choose 3 chairs from the green and blue chairs is C(2+4, 3) = C(6, 3) = 6! / (3! * (6-3)!) = 6! / (3! * 3!) = (6 * 5 * 4) / (3 * 2 * 1) = 20 ways.
Step 3: Calculate the number of ways to choose 3 chairs with at least one yellow chair
The number of ways to choose 3 chairs with at least one yellow chair is the total number of ways minus the number of ways calculated in step 2.
Number of ways = Total number of ways - Number of ways without yellow chair = 84 - 20 = 64 ways.
Therefore, the correct answer is option C) 64.
Step 1: Calculate the total number of ways to choose 3 chairs from all the available chairs
The total number of chairs in the room is 2 green chairs + 3 yellow chairs + 4 blue chairs = 9 chairs.
We need to choose 3 chairs from these 9 chairs, which can be done in C(9, 3) ways, where C(n, r) represents the number of combinations of choosing r items from a set of n items.
C(9, 3) = 9! / (3! * (9-3)!) = 9! / (3! * 6!) = (9 * 8 * 7) / (3 * 2 * 1) = 84 ways.
Step 2: Calculate the number of ways to choose 3 chairs without including any yellow chair
Since we want to find the number of ways to choose at least one yellow chair, we need to subtract the number of ways to choose 3 chairs without including any yellow chair from the total number of ways calculated in step 1.
To choose 3 chairs without including any yellow chair, we can choose from the green and blue chairs only.
The number of ways to choose 3 chairs from the green and blue chairs is C(2+4, 3) = C(6, 3) = 6! / (3! * (6-3)!) = 6! / (3! * 3!) = (6 * 5 * 4) / (3 * 2 * 1) = 20 ways.
Step 3: Calculate the number of ways to choose 3 chairs with at least one yellow chair
The number of ways to choose 3 chairs with at least one yellow chair is the total number of ways minus the number of ways calculated in step 2.
Number of ways = Total number of ways - Number of ways without yellow chair = 84 - 20 = 64 ways.
Therefore, the correct answer is option C) 64.
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