The Correct Answer and Explanation is:
The correct answer is (A).
Explanation
To find a simplified expression for the generalized binomial coefficient (1/4 choose n), we start with its definition:
(r choose k) = [r * (r-1) * (r-2) * … * (r-k+1)] / k!
For our specific case, r = 1/4 and k = n. Substituting these values, we get:
(1/4 choose n) = [ (1/4) * (1/4 – 1) * (1/4 – 2) * … * (1/4 – n + 1) ] / n!
Let’s simplify the product in the numerator. It consists of n terms. We can write each term as a fraction with a denominator of 4:
- 1/4
- 1/4 – 1 = (1 – 4)/4 = -3/4
- 1/4 – 2 = (1 – 8)/4 = -7/4
- 1/4 – 3 = (1 – 12)/4 = -11/4
- …
- 1/4 – (n-1) = [1 – 4(n-1)]/4 = (5 – 4n)/4
The product in the numerator is:
[1 * (-3) * (-7) * (-11) * … * (5 – 4n)] / 4^n
Now, let’s analyze the sign and magnitude of the numerator’s product. The first term, 1, is positive. All subsequent (n-1) terms are negative. The product of (n-1) negative numbers results in a sign of (-1)^(n-1).
Next, we consider the product of the absolute values of these numerators:
|1| * |-3| * |-7| * |-11| * … * |5 – 4n| = 1 * 3 * 7 * 11 * … * (4n-5)
(Note that for n ≥ 2, 5 – 4n is negative, so |5 – 4n| = -(5-4n) = 4n-5).
Combining the sign, the product of absolute values, and the denominator, we get:
(1/4 choose n) = [ (-1)^(n-1) * (1 * 3 * 7 * 11 * … * (4n-5)) ] / (4^n * n!)
This expression matches option (A). The product in option (A) is written as 3 * 7 * 11 * … * (4n-5). This is mathematically equivalent to 1 * 3 * 7 * 11 * … * (4n-5), as multiplying by 1 does not change the value. For the case n=1, the product is empty, which is defined as 1, correctly giving the numerator (-1)^0 * 1 = 1.
Therefore, the simplified expression is:
(A) [(-1)^(n-1) * 3 * 7 * 11 * … * (4n-5)] / (4^n * n!)
