On the refrigerator, mathematics is spelled out with $11$ magnets, one letter per magnet. two vowels and four consonants fall off and are put away in a bag. if the t's, m's, and a's are indistinguishable, how many distinct possible collections of letters could be put in the bag

ANSWER: 8 distinct possible collections

EXPLANATION:

Non extraneous information (Information that matters):
- Mathematics has 11 letters
- The bag must contain 4 vowels and 2 consonants
- t's, m's, and a's are indistinguishable
———
Bag Combinations Step 1:
List all:
aeaimt
aeaimh
aeaimm
aeaimc
aeaims
aeaith
aeaitt
aeaitc
aeaits
aeaihc
aeaihs
aeaics
———
Bag Combinations Step 2:
Change all t's, m's, and a's to another variable so they are indistinguishable (k) :
kekikk
kekikh
kekikk
kekikc
kekiks
kekikh
kekikk
kekitc
kekiks
kekihc
kekihs
kekics
———
Bag Combinations Step 3:
Remove doubles and list:
kekikk
kekikh
kekikc
kekiks
kekitc
kekihc
kekihs
kekics

i think there was a typo. you wrote pat brings 3 quarts of ice-cream to the party. each serving of ice-cream is 1 6 quart. how many total servings of ice-cream does pat bring to the party? i think the bold is supposed to be 1/6. so 1 quart of ice cream = 6 servings. 6 * 3 = 18.

the answer is 18.