It looks like you've uploaded an image of a math test on permutations, combinations, and probabilities. I'll transcribe and solve the two questions in the image. Here's the breakdown:

Q1:

Three players each cut one card from a standard deck. If order is important, in how many ways could they:

a) all hearts?
b) all aces?
c) all aces or hearts?

Solution for Q1:

1. All hearts (a):
   A deck has 13 hearts. Since order matters, this is a permutation problem: $P(13, 3) = \frac{13!}{(13-3)!} = \frac{13 \times 12 \times 11}{1} = 1716$

2. All aces (b):
   There are 4 aces in a deck. Permuting 3 cards out of 4: $P(4, 3) = \frac{4!}{(4-3)!} = \frac{4 \times 3 \times 2}{1} = 24$

3. All aces or hearts (c):
   The key is to calculate both cases and add them, while ensuring no double-counting.

   • Total hearts (as above): $P(13, 3) = 1716$
   • Total aces (as above): $P(4, 3) = 24$

   There’s no overlap between these two groups since they’re separate suits. Thus:
   $1716 + 24 = 1740$

Q2:

Match each scenario with its most likely subjective probability. Justify your answers.

Justifications:

• a) Canada winning a medal is highly likely since it has consistently happened in international events like the Olympics, so $P(A) = 0.9$.
• b) Random left-handed person: Left-handed people represent about 10% of the population, so $P(A) = 0.1$.
• c) Random high school student in grade 10: Assuming equal distribution among grades 9–12, each grade is roughly $\frac{1}{4}$, so $P(A) = 0.25$.

Final Answers:

Q1: a) 1716
b) 24
c) 1740

Q2:
a) 0.9
b) 0.1
c) 0.25

Would you like a detailed explanation of any step?

5 Questions to Expand Your Understanding:

1. How do permutations differ from combinations?
2. What changes if order isn’t important in Q1?
3. Can you explain the overlap principle for "all aces or hearts"?
4. How is subjective probability determined?
5. Can you create a similar problem with "order not being important"?

Tip:

Remember, permutations are used when order matters, and combinations are used when it doesn't!