part i 1 how many distinct arrangements of the word mathematics are th

Question

Part I
1. How many distinct arrangements of the word "MATHEMATICS" are there?
2. In a school, 12th grade students are enrolled in Mathematics and/or Computer Science.
If there are 300 12th grade students, and 100 are enrolled in Mathematics while 250
are enrolled in computer science, how many are enrolled in both Mathematics and
Computer Science?
3. Assuming a Five-Card Poker Hand is dealt:
How many of these hands contain exactly 3 diamonds?
• How many of these hands contain at least 3 diamonds?
• How many of these hands contain three black cards and two red cards?
4. In how many ways can we select a president, vice-president,
treasurer, and secretary from a group of 9 people? Assume that the same person
can hold more than one office.
5. A seven-person committee composed of Maya, Grace, Jonah, Zig, Esme, Miles, and
Tristan is to select a chairperson, secretary, and treasurer. How many selections are
there in which at least one office is held by Maya or Grace?
Assume that each person can hold at most one office.