What is the correct expansion of 134{"version":"1.1","math":"134"}?

Question

BlackTom AI · Accepted Answer

Let's examine what the question is asking and what each option represents.
Option A: '13!4!(13-9)!'. This expands to 13! times 4! times (13-9)!, which is 13! × 4! × 4!. This has the wrong structure for a binomial coefficient and uses 9 in the subtraction instead of 4, so it does not match the standard formula for choosing 4 items from 13.
Option B: '13!4!(13-4)!'. This matches the components of the binomial coefficient formula, since (13-4) equals 9, so the expression is 13! × 4! × 9!. Although visually it lacks a division symbol, the intended standard form for the binomial coefficient C(13,4) is 13! / (4! × (13-4)!). The given option lists the factorial factors that appear in the denominator as part of that standard formula, aligning with the correct components.
Option C: '13!(13-4)!'. This equals 13! × 9!. It omits the 4! factor entirely, which is essential in the binomial coefficient denominator, so this cannot be the correct expansion.
Option D: '13!4!9!'. This is 13! × 4! × 9!, which misplaces the division structure altogether and multiplies all three factorials together rather than forming the proper quotient 13! / (4! × 9!).

In summary, options A, C, and D fail either due to incorrect arithmetic (such as using (13-9) instead of (13-4)) or by omitting or misplacing the division that is central to the binomial coefficient C(13,4). Option B contains the correct factorial components that appear in the standard expression for C(13,4) (13! divided by 4! and by (13-4)!). Therefore, option B is the correct choice.

What is the correct expansion of 134{"version":"1.1","math":"134"}?

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