1. The problem is to evaluate the function $f(x) = \frac{\binom{4}{x}}{16}$ for $x = 0, 1, 2, 3, 4$ and understand its values. 2. The binomial coefficient $\binom{n}{k}$ represents the number of ways to choose $k$ elements from $n$ elements without regard to order. It is calculated as: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ 3. Here, $n=4$ and $k=x$. The denominator 16 is $2^4$, which is the total number of subsets of a 4-element set, making $f(x)$ a probability mass function of a binomial distribution with parameters $n=4$ and $p=\frac{1}{2}$. 4. Calculate each value: - $f(0) = \frac{\binom{4}{0}}{16} = \frac{1}{16}$ - $f(1) = \frac{\binom{4}{1}}{16} = \frac{4}{16}$ - $f(2) = \frac{\binom{4}{2}}{16} = \frac{6}{16}$ - $f(3) = \frac{\binom{4}{3}}{16} = \frac{4}{16}$ - $f(4) = \frac{\binom{4}{4}}{16} = \frac{1}{16}$ 5. These values sum to 1, confirming they form a valid probability distribution: $$\frac{1}{16} + \frac{4}{16} + \frac{6}{16} + \frac{4}{16} + \frac{1}{16} = \frac{16}{16} = 1$$ 6. The graph of $f(x)$ consists of discrete points at $x=0,1,2,3,4$ with heights equal to these probabilities, showing the binomial distribution shape symmetric around $x=2$. Final answer: The function values are $f(0)=\frac{1}{16}$, $f(1)=\frac{4}{16}$, $f(2)=\frac{6}{16}$, $f(3)=\frac{4}{16}$, and $f(4)=\frac{1}{16}$, representing a binomial probability mass function with $n=4$ and $p=\frac{1}{2}$.