Commerce Exam > Commerce Questions > If P (n,r) = C (n,r) thena)n = rb)r = 0 or 1c...
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If P (n,r) = C (n,r) then
- a)n = r
- b)r = 0 or 1
- c)r = 1 or n
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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If P (n,r) = C (n,r) thena)n = rb)r = 0 or 1c)r = 1 or nd)none of thes...
The correct answer is b) r = 0 or 1.
P (n,r) and C (n,r) are both mathematical notations that refer to the number of ways that a set of items can be arranged or selected. P (n,r) stands for the number of permutations of n items taken r at a time, and C (n,r) stands for the number of combinations of n items taken r at a time.
If P (n,r) = C (n,r), then it means that the number of permutations is equal to the number of combinations. This can only occur when r is equal to 0 or 1.
If r = 0, then there is only 1 way to select 0 items from a set of n items, which is to select nothing. If r = 1, then there are n ways to select 1 item from a set of n items (you can select any of the n items). In both of these cases, the number of permutations is equal to the number of combinations, as there is only 1 way to arrange 0 items and only 1 way to arrange 1 item.
Therefore, if P (n,r) = C (n,r), then r = 0 or 1.
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If P (n,r) = C (n,r) thena)n = rb)r = 0 or 1c)r = 1 or nd)none of thes...
Understanding P(n, r) and C(n, r)
In combinatorial mathematics, P(n, r) and C(n, r) represent different concepts:
- P(n, r): This denotes the number of permutations of n items taken r at a time. The formula is n! / (n - r)!.
- C(n, r): This denotes the number of combinations of n items taken r at a time. The formula is n! / [r! * (n - r)!].
Equating P(n, r) and C(n, r)
When we set P(n, r) equal to C(n, r), we have:
- n! / (n - r)! = n! / [r! * (n - r)!].
This simplifies to:
- r! = 1.
Implications of r! = 1
The factorial of a number r (r!) equals 1 in two specific cases:
- r = 0: By definition, 0! = 1.
- r = 1: Since 1! = 1.
Thus, the condition r! = 1 leads us to conclude that r can either be 0 or 1.
Conclusion
Given that P(n, r) = C(n, r) implies r must be 0 or 1, we can confirm that:
- The correct answer is option B: r = 0 or 1.
This insight clarifies the relationship between permutations and combinations under specific conditions.
In combinatorial mathematics, P(n, r) and C(n, r) represent different concepts:
- P(n, r): This denotes the number of permutations of n items taken r at a time. The formula is n! / (n - r)!.
- C(n, r): This denotes the number of combinations of n items taken r at a time. The formula is n! / [r! * (n - r)!].
Equating P(n, r) and C(n, r)
When we set P(n, r) equal to C(n, r), we have:
- n! / (n - r)! = n! / [r! * (n - r)!].
This simplifies to:
- r! = 1.
Implications of r! = 1
The factorial of a number r (r!) equals 1 in two specific cases:
- r = 0: By definition, 0! = 1.
- r = 1: Since 1! = 1.
Thus, the condition r! = 1 leads us to conclude that r can either be 0 or 1.
Conclusion
Given that P(n, r) = C(n, r) implies r must be 0 or 1, we can confirm that:
- The correct answer is option B: r = 0 or 1.
This insight clarifies the relationship between permutations and combinations under specific conditions.
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