Roster Form from Set-Builder Form

The governing concept is set-builder representation: state the common property of the elements and impose bounds that include exactly the listed values. Every element of K is a multiple of 4, and the multiples of 4 from 4 through 20 are 4, 8, 12, 16, and 20. Because both inequalities are inclusive, option A generates precisely K. Option B would give the factors of 20, including 1, 2, 5, and 10, which are not all in K. Option C concerns even digits, not all bounded multiples of 4, and option D describes numbers below 4, which excludes every member of K. Thus only option A has both the correct property and correct limits.

The governing result is the binomial theorem: (1 + x)¹⁰ = Σ ¹⁰Cᵣ xʳ. The term containing x⁴ is obtained by taking r = 4, so its coefficient is ¹⁰C₄. Equivalently, in the product of ten factors, choose x from exactly four factors and 1 from the other six; the four chosen positions can be selected in ¹⁰C₄ ways. Thus option B is correct. ¹⁰P₄ incorrectly treats the four positions as ordered, 10⁴ does not represent the required selection, and 4! counts arrangements among selected positions even though multiplication is commutative and order does not create new terms.