In the image, there are three mathematical problems related to permutations and combinations. Let's solve each part one by one.

1. a. $\frac{10! \times 6!}{5! \times 7!}$

   We simplify the factorial expressions step by step:

   • $10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5!$
   • $6! = 6 \times 5!$
   • $7! = 7 \times 6!$

   Now we substitute these into the expression:

   $\frac{10! \times 6!}{5! \times 7!} = \frac{(10 \times 9 \times 8 \times 7 \times 6 \times 5!) \times (6 \times 5!)}{5! \times (7 \times 6!)}$

   Cancel out the common terms $5!$ and $6!$:

   $= \frac{10 \times 9 \times 8 \times 7 \times 6 \times 6}{7 \times 6}$

   Now cancel the common $7 \times 6$:

   $= 10 \times 9 \times 8 \times 6 = 4320$

   So, the answer to part a is 4320.

2. b. $13P4$

   This is a permutation problem where we use the permutation formula $nP r = \frac{n!}{(n-r)!}$.

   For $13P4$, it becomes:

   $13P4 = \frac{13!}{(13-4)!} = \frac{13!}{9!}$

   We simplify by cancelling $9!$ from the numerator and denominator:

   $13P4 = 13 \times 12 \times 11 \times 10 = 17160$

   So, the answer to part b is 17160.

3. c. $20C3$

   This is a combination problem, and we use the combination formula $nC r = \frac{n!}{r! \times (n-r)!}$.

   For $20C3$, it becomes:

   $20C3 = \frac{20!}{3! \times (20-3)!} = \frac{20!}{3! \times 17!}$

   Simplifying by cancelling $17!$:

   $20C3 = \frac{20 \times 19 \times 18}{3!} = \frac{20 \times 19 \times 18}{6}$

   Now, calculate the result:

   $= \frac{6840}{6} = 1140$

   So, the answer to part c is 1140.

Summary of answers:

• a. $\frac{10! \times 6!}{5! \times 7!} = 4320$
• b. $13P4 = 17160$
• c. $20C3 = 1140$

Would you like further details on any of these steps or have more questions on permutations and combinations?

Here are some related questions you might find helpful:

1. How do you differentiate between permutations and combinations?
2. What is the difference between $n!$ and $nC r$?
3. How do you calculate the factorial of a large number?
4. What are real-life applications of permutations and combinations?
5. How do you simplify complex factorial expressions?

Tip: When simplifying factorial expressions, always look for terms that cancel out, such as $5!$ or $6!$, to make the calculations much easier!