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If six times the number of permutations of n objects taken 3 at a time is equal to seven times the number of permutations of (n-1) objects taken 3 at a time, find the value of n.
- a)n = 20
- b)n = 19
- c)n = 21
- d)n = 18
Correct answer is option 'C'. Can you explain this answer?
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If six times the number of permutations of n objects taken 3 at a time...
Most Upvoted Answer
If six times the number of permutations of n objects taken 3 at a time...
Given:
- Six times the number of permutations of n objects taken 3 at a time = 6Pₙ₃
- Seven times the number of permutations of (n-1) objects taken 3 at a time = 7Pₙ₋₁₃
To Find: The value of n
Solution:
Step 1: Write the formula for permutations
nPₓ = n!/(n-x)!
Where n is the total number of objects and x is the number of objects taken at a time.
Step 2: Substitute the given values in the formula
6Pₙ₃ = 6n!/(n-3)!
7Pₙ₋₁₃ = 7(n-1)!/(n-4)!
Step 3: Equate the two expressions given in the question
6n!/(n-3)! = 7(n-1)!/(n-4)!
Step 4: Simplify the equation
6n(n-1)(n-2) = 7(n-1)(n-2)(n-3)
Step 5: Cancel out common factors on both sides
6n = 7(n-3)
6n = 7n - 21
n = 21
Therefore, the value of n is 21.
Answer: Option (c)
- Six times the number of permutations of n objects taken 3 at a time = 6Pₙ₃
- Seven times the number of permutations of (n-1) objects taken 3 at a time = 7Pₙ₋₁₃
To Find: The value of n
Solution:
Step 1: Write the formula for permutations
nPₓ = n!/(n-x)!
Where n is the total number of objects and x is the number of objects taken at a time.
Step 2: Substitute the given values in the formula
6Pₙ₃ = 6n!/(n-3)!
7Pₙ₋₁₃ = 7(n-1)!/(n-4)!
Step 3: Equate the two expressions given in the question
6n!/(n-3)! = 7(n-1)!/(n-4)!
Step 4: Simplify the equation
6n(n-1)(n-2) = 7(n-1)(n-2)(n-3)
Step 5: Cancel out common factors on both sides
6n = 7(n-3)
6n = 7n - 21
n = 21
Therefore, the value of n is 21.
Answer: Option (c)
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Community Answer
If six times the number of permutations of n objects taken 3 at a time...
Number of permutation of n things taken r at a time =nPr
number of permutation of n things taken 3 at a time=nP3
number of permutation of (n-1) thing taken 3 at a time= n-1P3
We are given that
6*nP3=7*n-1P3
we know that
nPr=n!/(n-r)!
so,
6*(n! /(n-3)!)=7*((n-1)!/(n-4)!)
6*((n)(n-1)(n-2)(n-3)!/(n-3)!)=7*((n-1)(n-2)(n-3)(n-4)!/(n-4)!)
cancel (n-3)! &(n-4)! from LHS and RHS respectively
6*(n) (n-1)(n-2)=7*(n-1)(n-2)(n-3)
then cancel (n-1)(n-2) by both sides
So,
6(n)=7(n-3)
6n=7n-21
7n-6n=21
n=21
number of permutation of n things taken 3 at a time=nP3
number of permutation of (n-1) thing taken 3 at a time= n-1P3
We are given that
6*nP3=7*n-1P3
we know that
nPr=n!/(n-r)!
so,
6*(n! /(n-3)!)=7*((n-1)!/(n-4)!)
6*((n)(n-1)(n-2)(n-3)!/(n-3)!)=7*((n-1)(n-2)(n-3)(n-4)!/(n-4)!)
cancel (n-3)! &(n-4)! from LHS and RHS respectively
6*(n) (n-1)(n-2)=7*(n-1)(n-2)(n-3)
then cancel (n-1)(n-2) by both sides
So,
6(n)=7(n-3)
6n=7n-21
7n-6n=21
n=21
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If six times the number of permutations of n objects taken 3 at a time is equal to seven times the number of permutations of (n-1) objects taken 3 at a time, find the value of n.a)n = 20b)n = 19c)n = 21d)n = 18Correct answer is option 'C'. Can you explain this answer?
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