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In a college there are 20 professors including the principal and the vice principal. A committee of 5 is to be formed. In how many ways it can be formed so that neither the principal nor the vice principal is included?
- a)C(18,2)
- b)C(20,2)
- c)C(20,18)
- d)C(18,5)
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
In a college there are 20 professors including the principal and the v...
Out of 20 professors, excluding the principal and vice-principal, 18 professors are left.
No. of ways of selecting 5 professors out of 18 = 18C5
No. of ways of selecting 5 professors out of 18 = 18C5
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In a college there are 20 professors including the principal and the v...
To form a committee of 5 members from a total of 20 professors, including the principal and vice principal, such that neither the principal nor the vice principal is included, we need to subtract the number of ways in which the committee can be formed with either the principal or the vice principal from the total number of ways in which the committee can be formed.
Total Ways to Form a Committee:
To calculate the total number of ways to form a committee of 5 members from the 20 professors, we can use the combination formula denoted by C(n, r), where n is the total number of professors and r is the number of professors we want to choose for the committee.
Total ways = C(20, 5) = 20! / (5! * (20-5)!)
Ways to Include Either Principal or Vice Principal:
To calculate the number of ways to form a committee that includes either the principal or the vice principal, we need to consider two cases:
1. Including the principal: In this case, we need to choose 4 more professors from the remaining 19 professors (excluding the vice principal).
Ways to include the principal = C(19, 4)
2. Including the vice principal: In this case, we also need to choose 4 more professors from the remaining 19 professors (excluding the principal).
Ways to include the vice principal = C(19, 4)
Since we want to exclude both the principal and the vice principal, we need to subtract the total ways of including either the principal or the vice principal from the total number of ways.
Number of ways to form a committee excluding the principal and vice principal:
Total ways - (Ways to include the principal + Ways to include the vice principal)
= C(20, 5) - ( C(19, 4) + C(19, 4) )
= C(18, 5)
Therefore, the correct answer is option 'D', C(18, 5), which represents the number of ways to form a committee of 5 members excluding both the principal and the vice principal.
Total Ways to Form a Committee:
To calculate the total number of ways to form a committee of 5 members from the 20 professors, we can use the combination formula denoted by C(n, r), where n is the total number of professors and r is the number of professors we want to choose for the committee.
Total ways = C(20, 5) = 20! / (5! * (20-5)!)
Ways to Include Either Principal or Vice Principal:
To calculate the number of ways to form a committee that includes either the principal or the vice principal, we need to consider two cases:
1. Including the principal: In this case, we need to choose 4 more professors from the remaining 19 professors (excluding the vice principal).
Ways to include the principal = C(19, 4)
2. Including the vice principal: In this case, we also need to choose 4 more professors from the remaining 19 professors (excluding the principal).
Ways to include the vice principal = C(19, 4)
Since we want to exclude both the principal and the vice principal, we need to subtract the total ways of including either the principal or the vice principal from the total number of ways.
Number of ways to form a committee excluding the principal and vice principal:
Total ways - (Ways to include the principal + Ways to include the vice principal)
= C(20, 5) - ( C(19, 4) + C(19, 4) )
= C(18, 5)
Therefore, the correct answer is option 'D', C(18, 5), which represents the number of ways to form a committee of 5 members excluding both the principal and the vice principal.
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In a college there are 20 professors including the principal and the vice principal. A committee of 5 is to be formed. In how many ways it can be formed so that neither the principal nor the vice principal is included?a)C(18,2)b)C(20,2)c)C(20,18)d)C(18,5)Correct answer is option 'D'. Can you explain this answer?
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