The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to :

We need to find the number of ways to choose 5 alphabets from the word MATHEMATICS (not necessarily distinct).

The distinct letters in MATHEMATICS with their frequencies are:

M: 2, A: 2, T: 2, H: 1, E: 1, I: 1, C: 1, S: 1

So we have 8 distinct letters, with 3 letters (M, A, T) appearing twice and 5 letters (H, E, I, C, S) appearing once.

We need to choose 5 letters where the chosen set respects the available multiplicities. We consider cases based on the pattern of repetitions.

Case 1: All 5 distinct.

Choose 5 from 8 distinct letters: $$\binom{8}{5} = 56$$.

Case 2: One pair + 3 distinct.

Choose which letter is repeated (from M, A, T): $$\binom{3}{1} = 3$$ ways.

Choose remaining 3 distinct letters from the other 7: $$\binom{7}{3} = 35$$ ways.

Total: $$3 \times 35 = 105$$.

Case 3: Two pairs + 1 distinct.

Choose 2 letters to be repeated (from M, A, T): $$\binom{3}{2} = 3$$ ways.

Choose 1 distinct letter from the remaining 6: $$\binom{6}{1} = 6$$ ways.

Total: $$3 \times 6 = 18$$.

Case 4: Three or more of the same letter.

Not possible since no letter appears more than twice.

Grand total = $$56 + 105 + 18 = 179$$.

The correct answer is Option 1: 179.