Explanation:

We are given that $a, b, c, d, e$ are natural numbers (positive integers) and their sum is $a+b+c+d+e=10$. We need to find the maximum value of their product $P = a \times b \times c \times d \times e$. \nFor a fixed sum of positive numbers, their product is maximized when the numbers are as close to each other as possible. This is a fundamental principle related to the AM-GM inequality.\nIn this case, we have $5$ numbers whose sum is $10$. The average value of these numbers is $\frac{10}{5} = 2$. \nIf we set all the numbers to be equal to their average, i.e., $a=b=c=d=e=2$, then:\nSum: $2+2+2+2+2 = 10$. This satisfies the given condition.\nProduct: $2 \times 2 \times 2 \times 2 \times 2 = 2^5 = 32$. \nSince $2$ is a natural number, this distribution is valid. Any other distribution of natural numbers that sums to $10$ will result in a smaller product. For example:\nIf we choose $1, 2, 2, 2, 3$ (sum is $10$): Product $= 1 \times 2 \times 2 \times 2 \times 3 = 24$.\nIf we choose $1, 1, 2, 3, 3$ (sum is $10$): Product $= 1 \times 1 \times 2 \times 3 \times 3 = 18$.\nBoth $24$ and $18$ are less than $32$. \nLet's evaluate the options:\nOption (1) $36$: This value is not achievable for 5 natural numbers summing to $10$.\nOption (2) $30$: This value is less than $32$.\nOption (3) $32$: This matches our calculated maximum value.\nOption (4) $48$: This value is not achievable for 5 natural numbers summing to $10$. \nTherefore, the maximum value of $a \times b \times c \times d \times e$ is $32$. The correct option is (3) 32.