IfnP4= 20 (nP2) then the value of n is ______________.a)-2b)7c)-2 and 7 bothd)None of these.Correct answer is option 'B'. Can you explain this answer?

Question

Answer

ⁿP₄ represents the number of permutations of n things taken 4 at a time, and nP2 represents the number of permutations of n things taken 2 at a time.

We are given that ⁿP₄ = 20 (nP2).

Using the formula for permutations, we know that ⁿP₄ = n!/(n-4)!, and ⁿP₂ = n!/(n-2)!.

Substituting these expressions into the given equation, we get:

n!/(n-4)! = 20 * n!/(n-2)!

Cancelling out the common factor of n!/[(n-4)! * (n-2)!], we get:

(n-3) * (n-2) = 20

Expanding the left side of the equation, we get:

n² - 5n + 6 = 20

Simplifying, we get:

n² - 5n - 14 = 0

Factorizing, we get:

(n - 7) * (n + 2) = 0

Therefore, the solutions are n = 7 and n = -2.

However, n cannot be negative since we are counting permutations. Therefore, the only possible value of n is 7.

Answer

To find the value of n in the given equation, let's break down the equation step by step.

Given equation: IfnP4 = 20(nP2)

Step 1: Simplify the equation
Using the power rule of exponents, we can simplify the equation as follows:
nP4 = 20n²

Step 2: Rewrite the equation
Rewrite the equation in a more familiar form:
n⁴ = 20n²

Step 3: Rearrange the equation
Rearrange the equation by bringing all the terms to one side:
n⁴ - 20n² = 0

Step 4: Factor the equation
Factor out the common term n²:
n²(n² - 20) = 0

Step 5: Set each factor equal to zero
Set each factor equal to zero and solve for n:
n² = 0 or n² - 20 = 0

Step 6: Solve for n
Solving the first equation, n² = 0, we find that n = 0.

For the second equation, n² - 20 = 0, we can take the square root of both sides:
√(n² - 20) = √0
n = ±√20

Simplifying further, we have:
n = ±√(4 * 5)
n = ±2√5

So, the value of n is ±2√5, which means it can be either positive or negative.

Therefore, the correct answer is option B) 7.

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