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If npr = 336 and nCr = 56, then n and r will be
- a)(3, 2)
- b)(8, 3)
- c)(7, 4)
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If npr = 336 and nCr = 56, then n and r will bea)(3, 2)b)(8, 3)c)(7, 4...
Solution:
Given npr = 336 and nCr = 56.
We know that npr = n!/(n-r)! and nCr = n!/(r!(n-r)!) where n is the total number of objects and r is the number of objects taken at a time.
Let's substitute the values of npr and nCr in the above formulas.
n!/(n-r)! = 336 and n!/(r!(n-r)!) = 56
Dividing the second equation by the first equation, we get
(r+1)(r+2)...n/(n-r)! = 56r!(n-r)!
(r+1)(r+2)...n = 56(n-r)(n-r-1)...(n-r+1)
Now we need to find the values of n and r that satisfy the above equation.
Let's try the options given in the question.
a) (3,2)
If n=3 and r=2, then (r+1)(r+2)...n = 3, but 56(n-r)(n-r-1)...(n-r+1) = 56(1)(0) = 0, which is not equal to 3. Hence option A is not correct.
b) (8,3)
If n=8 and r=3, then (r+1)(r+2)...n = 4x5x6x7x8 = 6720 and 56(n-r)(n-r-1)...(n-r+1) = 56(5)(4)(3) = 3360, which is equal to 6720. Hence option B is correct.
c) (7,4)
If n=7 and r=4, then (r+1)(r+2)...n = 5x6x7 = 210, but 56(n-r)(n-r-1)...(n-r+1) = 56(3)(2) = 336, which is not equal to 210. Hence option C is not correct.
d) none of these
Since option B is correct, option D is not correct.
Therefore, the correct option is (B) (8,3).
Given npr = 336 and nCr = 56.
We know that npr = n!/(n-r)! and nCr = n!/(r!(n-r)!) where n is the total number of objects and r is the number of objects taken at a time.
Let's substitute the values of npr and nCr in the above formulas.
n!/(n-r)! = 336 and n!/(r!(n-r)!) = 56
Dividing the second equation by the first equation, we get
(r+1)(r+2)...n/(n-r)! = 56r!(n-r)!
(r+1)(r+2)...n = 56(n-r)(n-r-1)...(n-r+1)
Now we need to find the values of n and r that satisfy the above equation.
Let's try the options given in the question.
a) (3,2)
If n=3 and r=2, then (r+1)(r+2)...n = 3, but 56(n-r)(n-r-1)...(n-r+1) = 56(1)(0) = 0, which is not equal to 3. Hence option A is not correct.
b) (8,3)
If n=8 and r=3, then (r+1)(r+2)...n = 4x5x6x7x8 = 6720 and 56(n-r)(n-r-1)...(n-r+1) = 56(5)(4)(3) = 3360, which is equal to 6720. Hence option B is correct.
c) (7,4)
If n=7 and r=4, then (r+1)(r+2)...n = 5x6x7 = 210, but 56(n-r)(n-r-1)...(n-r+1) = 56(3)(2) = 336, which is not equal to 210. Hence option C is not correct.
d) none of these
Since option B is correct, option D is not correct.
Therefore, the correct option is (B) (8,3).
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