Since the product of the two integers should be negative, we need to select a positive integer from one set and a negative integer from the other set.

Since set M has all terms negative, we need to select the positive integer from set T.

Set T has (n + 3) terms, of which, n are positive.

Number of ways of selecting a negative integer from set \(M = C^5_1 = 5 \)

Number of ways of selecting a positive integer from set \(T =C^n_1 =n\)

Thus, number of favorable cases = 5 x n.

Total number of cases = (# of ways of selecting an integer from set M) x (# of ways of selecting an integer from set T)


\(= C^5_1 \times C^{n+3}_1 = 5 x (n + 3) \)

Thus, required probability


Number of favorable cases/Total number of cases

\(\frac{5n}{5(n+3)}\)

\(\frac{n}{n+3}\)

Thus, we have:

\(\frac{n}{n+3}>\frac{3}{5}\)

\(5n>3n+9\)

\(n>\frac{9}{2}=4.5\)

Among the options, the eligible values for n are 5, 7, 8, and 9.
The correct answers are options B, C, D and E.