Question

Find the value of n! (n - r)!r! when n = r.

Answer 1

When n is equal to r, the expression n!(n - r)!r! simplifies to (n!)^2 which is the factorial of n raised to the power of two.

Step-by-step explanation:

When n = r, the expression n! / (n - r)! r! simplifies because (n - r)! and r! become 0! (which is equal to 1) since n and r are equal. Therefore, the expression becomes:

(n−r)!⋅r!/n!

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The original question asks, 'Find the value of n! (n - r)!r! when n = r.' Given that n is equal to r, it simplifies the equation substantially. When n and r are equal, (n - r)! becomes 0!, which is 1, and similarly n! and r! will be the same value, hence multiplying the same number.

By this logic, the formula simplifies down to n! × 1 × n!, which is essentially (n!)2.

For example, if n and r both equal 3, then the calculation would be 3! × (3 - 3)! × 3!, which is 6 × 1 × 6 = 36 or (6)2. Hence the value of n!(n - r)!r! when n = r is the square of n!.

Answer 2

The value of n! (n - r)! r! when n = r simplifies to (n!)^2, because (n - r)! equals 1 when n equals r.

Step-by-step explanation:

The question asks to find the value of n! (n - r)! r! when n = r. In mathematics, when we have a factorial expression like n!, it means we multiply all integers from 1 to n inclusive. If n = r, then n! (n - r)! r! simplifies significantly.

Since r is equal to n, the expression (n - r)! becomes (n - n)!, which is 0!. By definition, 0! is equal to 1. Therefore, the final value of the expression when n = r is simply n! × 1 × r! or (n!)^2 because r! is the same as n!.

For example, if n = 3 and thus r = 3, we would calculate 3! × (3 - 3)! × 3! which is 3! × 0! × 3! or (3!)^2. This equals 6 × 1 × 6, which is 36.