(iii) To find the number of selections with exactly 1 M and exactly 1 E, we first choose 1 M from the 2 Ms and 1 E from the 2 Es. This leaves us with 5 other letters (I, N, C, A, T) to choose 3 more letters from. The number of ways to choose 3 letters from these 5 is given by:

\(\binom{5}{3} = 10\)

(iv) To find the number of selections with at least 1 M and at least 1 E, we consider two methods:

Method 1: Consider separate groups:

• MME: Choose 2 Ms and 1 E, then choose 2 more letters from the remaining 5 letters (I, N, C, A, T): \(\binom{5}{2} = 10\)
• MEE: Choose 1 M and 2 Es, then choose 2 more letters from the remaining 5 letters: \(\binom{5}{2} = 10\)
• MMEE: Choose 2 Ms and 2 Es, then choose 1 more letter from the remaining 5 letters: \(\binom{5}{1} = 5\)

Summing these scenarios gives: \(10 + 10 + 5 = 25\)

Adding the scenario from part (iii) ME***: \(10\) (as calculated in part iii)

\(Total = 35\)

Method 2: Consider the criteria are met if ME are chosen:

Choose ME, then choose 3 more letters from the remaining 7 letters (I, N, C, A, T, M, E):

\(\binom{7}{3} = 35\)

Thus, the total number of selections is 35.