1. ## Binomial Coefficient

Hi!

I'm preparing for a midterm next week, and am going over the exercises in my textbook. I think this is correct, but just want to double check my understanding. Appreciate your help!

Question: Mixed Matched Marvin has a drawer full of 30 different socks (no two are the same). Ho reaches in and grabs two. In how many different ways can he do this? Now he puts them on his feet (presumably, one of the left and the other on the right). In how many different ways can he do that?

The answer to the first question seems to be $\displaystyle 30 \choose 2$, or $\displaystyle \frac{30!}{2!(30-2)!}$.

The second question only seems to multiply the first number of choices by two. That is,

$\displaystyle \frac{2 \times 30!}{2!(30-2)!}$

Simplified, this is:

$\displaystyle 30_2$ or $\displaystyle 30 \times 29$

Is this correct? Thanks!

2. Originally Posted by absvalue
Hi!

I'm preparing for a midterm next week, and am going over the exercises in my textbook. I think this is correct, but just want to double check my understanding. Appreciate your help!

Question: Mixed Matched Marvin has a drawer full of 30 different socks (no two are the same). Ho reaches in and grabs two. In how many different ways can he do this? Now he puts them on his feet (presumably, one of the left and the other on the right). In how many different ways can he do that?

The answer to the first question seems to be $\displaystyle 30 \choose 2$, or $\displaystyle \frac{30!}{2!(30-2)!}$.

The second question only seems to multiply the first number of choices by two. That is,

$\displaystyle \frac{2 \times 30!}{2!(30-2)!}$

Simplified, this is:

$\displaystyle 30_2$ or $\displaystyle 30 \times 29$

Is this correct? Thanks!
Right.

Another way to look at the second part, where the order of the socks matters, is that it is the number of permutations of 30 objects taken 2 at a time.