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From a group of persons, the number of ways of selecting 5 persons is equal to that of 8 persons. The number of persons in the group is
- a)13
- b)40
- c)18
- d)21
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
From a group of persons, the number of ways of selecting 5 persons is ...
r = 5 & n - r = 8 ⇒ n = 13
Most Upvoted Answer
From a group of persons, the number of ways of selecting 5 persons is ...
Understanding the Problem
In this problem, we need to find the total number of persons in a group where the number of ways to select 5 persons is equal to the number of ways to select 8 persons.
Combination Formula
The number of ways to choose r persons from n persons is given by the combination formula:
- C(n, r) = n! / [r! * (n - r)!]
Where "!" denotes factorial, meaning the product of all positive integers up to that number.
Setting Up the Equation
According to the problem, we set up the equation based on the combination formula:
- C(n, 5) = C(n, 8)
This translates to:
- n! / [5! * (n - 5)!] = n! / [8! * (n - 8)!]
Simplifying the Equation
Since n! appears in both sides of the equation, we can cancel it out:
- 1 / [5! * (n - 5)!] = 1 / [8! * (n - 8)!]
Next, we can cross-multiply to simplify further:
- 8! * (n - 8)! = 5! * (n - 5)!
Using Factorial Relationships
We know:
- 8! = 8 * 7 * 6 * 5!
- So, replacing 8! in the equation, we get:
- (8 * 7 * 6) * (n - 8)! = (n - 5) * (n - 6) * (n - 7) * (n - 8)!
This leads us to understand the relationship between n and the factorials.
Finding the Value of n
By solving the simplified equation, we find:
- n - 5 = 8 * 7 * 6
- n = 21
Thus, the number of persons in the group is 21.
Conclusion
The correct answer to the problem is option 'A', which indicates that there are 21 persons in the group.
In this problem, we need to find the total number of persons in a group where the number of ways to select 5 persons is equal to the number of ways to select 8 persons.
Combination Formula
The number of ways to choose r persons from n persons is given by the combination formula:
- C(n, r) = n! / [r! * (n - r)!]
Where "!" denotes factorial, meaning the product of all positive integers up to that number.
Setting Up the Equation
According to the problem, we set up the equation based on the combination formula:
- C(n, 5) = C(n, 8)
This translates to:
- n! / [5! * (n - 5)!] = n! / [8! * (n - 8)!]
Simplifying the Equation
Since n! appears in both sides of the equation, we can cancel it out:
- 1 / [5! * (n - 5)!] = 1 / [8! * (n - 8)!]
Next, we can cross-multiply to simplify further:
- 8! * (n - 8)! = 5! * (n - 5)!
Using Factorial Relationships
We know:
- 8! = 8 * 7 * 6 * 5!
- So, replacing 8! in the equation, we get:
- (8 * 7 * 6) * (n - 8)! = (n - 5) * (n - 6) * (n - 7) * (n - 8)!
This leads us to understand the relationship between n and the factorials.
Finding the Value of n
By solving the simplified equation, we find:
- n - 5 = 8 * 7 * 6
- n = 21
Thus, the number of persons in the group is 21.
Conclusion
The correct answer to the problem is option 'A', which indicates that there are 21 persons in the group.
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From a group of persons, the number of ways of selecting 5 persons is equal to that of 8 persons. The number of persons in the group isa)13b)40c)18d)21Correct answer is option 'A'. Can you explain this answer?
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