[math: 42C8]^{42}C_8 equals

Question

BlackTom AI · Accepted Answer

To determine 42C8, recall the definition of a binomial coefficient: nCk = n! / (k!(n−k)!). In this problem, n = 42 and k = 8, so 42C8 = 42! / (8! * 34!).

Option a appears to be the reciprocal of the correct form, since it shows 34!8! / 42!, which is equal to 1 / (42! / (34!8!)) = 1 / (42C8). This is not the same as 42C8, so it is incorrect.

Option b shows something like 42! / 8! (and possibly missing the 34! in the denominator or with an unclear grouping). Without the necessary division by 34! as well, this expression does not equal 42C8, so it cannot be correct.

Option c presents 42! / 34!, which would imply canceling the 8! in the denominator entirely and leaving a ratio of factorials that does not match the binomial coefficient formula. This is not how 42C8 is defined, so it is incorrect.

Option d correctly matches the binomial coefficient formula: 42! / (34! 8!). The order of multiplication in the denominator is commutative, so 34!8! is the same as 8!34!, and this exactly implements nCk = n! / (k!(n−k)!). This is the valid representation of 42C8.

Option e, if present, would presumably be 42!, which ignores the necessary division by 8! and 34!, so it cannot be equal to 42C8.

In summary, only the expression with 42! in the numerator and both 34! and 8! in the denominator matches the standard binomial coefficient formula for 42C8; all other forms fail to include the required factorials in the correct arrangement or represent reciprocals or overly simplified versions that do not yield the same value.

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