1. **Problem statement:** A discrete random variable $X$ has a probability mass function (pmf) given by $f(x) = c(x + 1)$ for $x = 0, 1, 2, 3$. We need to find the constant $c$. 2. **Formula and rules:** The sum of all probabilities for a pmf must equal 1: $$\sum_x f(x) = 1$$ Since $f(x) = c(x+1)$, we have: $$c(0+1) + c(1+1) + c(2+1) + c(3+1) = 1$$ 3. **Calculate the sum:** $$c(1) + c(2) + c(3) + c(4) = c(1+2+3+4) = c \times 10 = 1$$ 4. **Solve for $c$:** $$c = \frac{1}{10}$$ 5. **Interpretation:** The constant $c$ is $\frac{1}{10}$ to ensure the probabilities sum to 1. **Final answer:** $$c = \frac{1}{10}$$