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If nPr = nPr + 1 and nCr = nCr – 1, then the values of n and r are
- a)4, 3
- b)3, 2
- c)4, 2
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If nPr = nPr + 1 and nCr = nCr – 1, then the values of n and r area)4...
To find the values of n and r, we need to understand the properties of nPr and nCr.
Properties of nPr:
1. nPr represents the number of permutations of r objects taken from a set of n objects.
2. The formula for nPr is n!/(n-r)!, where n! represents the factorial of n.
3. The value of nPr is always greater than or equal to n.
Properties of nCr:
1. nCr represents the number of combinations of r objects taken from a set of n objects.
2. The formula for nCr is n!/(r!(n-r)!).
3. The value of nCr is always greater than or equal to 1.
Given that nPr = nPr - 1 and nCr = nCr - 1, we can make the following observations:
1. nPr = nPr - 1:
- From the properties of nPr, we know that nPr is always greater than or equal to n.
- Therefore, nPr - 1 should be equal to n.
- This implies that nPr should be equal to n + 1.
2. nCr = nCr - 1:
- From the properties of nCr, we know that nCr is always greater than or equal to 1.
- Therefore, nCr - 1 should be equal to 1.
- This implies that nCr should be equal to 2.
Based on the above observations, we can solve for the values of n and r:
1. nPr = nPr - 1:
- nPr = n + 1
- nPr - 1 = n
- Therefore, n + 1 = n
- This is not possible as it contradicts the properties of nPr.
- Hence, this equation has no solution.
2. nCr = nCr - 1:
- nCr = 2
- nCr - 1 = 1
- Therefore, 2 = 1
- This is not possible as it contradicts the properties of nCr.
- Hence, this equation has no solution.
Therefore, there are no values of n and r that satisfy both equations simultaneously. Hence, the correct answer is option 'D' - None of these.
Properties of nPr:
1. nPr represents the number of permutations of r objects taken from a set of n objects.
2. The formula for nPr is n!/(n-r)!, where n! represents the factorial of n.
3. The value of nPr is always greater than or equal to n.
Properties of nCr:
1. nCr represents the number of combinations of r objects taken from a set of n objects.
2. The formula for nCr is n!/(r!(n-r)!).
3. The value of nCr is always greater than or equal to 1.
Given that nPr = nPr - 1 and nCr = nCr - 1, we can make the following observations:
1. nPr = nPr - 1:
- From the properties of nPr, we know that nPr is always greater than or equal to n.
- Therefore, nPr - 1 should be equal to n.
- This implies that nPr should be equal to n + 1.
2. nCr = nCr - 1:
- From the properties of nCr, we know that nCr is always greater than or equal to 1.
- Therefore, nCr - 1 should be equal to 1.
- This implies that nCr should be equal to 2.
Based on the above observations, we can solve for the values of n and r:
1. nPr = nPr - 1:
- nPr = n + 1
- nPr - 1 = n
- Therefore, n + 1 = n
- This is not possible as it contradicts the properties of nPr.
- Hence, this equation has no solution.
2. nCr = nCr - 1:
- nCr = 2
- nCr - 1 = 1
- Therefore, 2 = 1
- This is not possible as it contradicts the properties of nCr.
- Hence, this equation has no solution.
Therefore, there are no values of n and r that satisfy both equations simultaneously. Hence, the correct answer is option 'D' - None of these.
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Community Answer
If nPr = nPr + 1 and nCr = nCr – 1, then the values of n and r area)4...
We have nPr = nPr+1
Or n - r = 1 …(1)
Also, nCr - nCr-1 → r + r - 1 = n
→ 2r - n = 1 …(2)
Solving (1) and (2), we get r = 2 and n = 3
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