In how many ways can three letters (not necessarily distinct from one another) be selected from the letters of the word PARALLELOGRAMS?

First, let's count the frequency of each letter in PARALLELOGRAMS:

Letter	Frequency
A	3
L	3
R	2
P, E, O, G, M, S	1 each

Total distinct letters = 9

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Since we're selecting 3 letters (not necessarily distinct), we have three possible scenarios:

All three letters are identical

Only letters appearing at least 3 times can be selected this way: A and L

Ways = 2 (AAA, LLL)

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Two letters are identical, one is different

Letters that can appear twice: A, R, L (since they appear at least twice)

• Two A's + one other letter → The other can be any of the remaining 8 distinct letters (P, R, L, E, O, G, M, S) → 8 ways
• Two R's + one other letter → 8 ways
• Two L's + one other letter → 8 ways

Total = 8 + 8 + 8 = 24

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All three letters are different

We need to choose 3 different letters from 9 distinct letters.

Using combinations: $C(9,3) = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = \frac{504}{6} = 84$

Ways = 84

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Total number of ways = 2 + 24 + 84 = 110