1.

I was wondering if I solved this correctly: $\displaystyle 13x \equiv 1 (mod 28)$

gcd(28,13) = 13 * 2 + 2

13 = 6 * 2 + 1

so:

$\displaystyle 1 = 13 - (6 * 2)$

$\displaystyle 1 = 13 - (6 * (28 - 13*2))$

$\displaystyle 1 = 13 - (6*28 - 13*(2*6)) = 13 - (6*28 - 13*12)$

$\displaystyle 1 = 13 - 6*28 + 13*12$

$\displaystyle 1 = 13*13 - 6*28$

$\displaystyle 13*13 \equiv 1 (mod 28)$

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2.

You have a bag of pink, yellow and green balls (woot).

You have to pick 14 balls to be sure of at least 2 pink.

You have to pick 12 balls to be sure of at least 2 yellow.

You have to pick 8 balls to be sure of at least 2 green.

How many of each?

I thought of this as:

yellow+green = 12

pink+green = 10

pink+yellow = 6

Add them all together: 2*pink + 2*green + 2*yellow = 28. Divide by 2, and you get 14 balls total.

If yellow+green = 12, then there are 2 pink.

If pink+green = 10, then there are 4 yellow.

If pink+yellow = 6, then there are 8 green.

Did I do this correctly?

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3.

An academic department rates its graduate program applicants numerically. Six different factors are taken into account. These different factors are weighted differently, with the integral weights chosen to add up to 100. The weight for each individual factor is at least 10. -> In how many ways could the department choose the weights?

Each department has at least 10, so you're distributing 100-(6*10) = 40 points among six factors, however, the answer key I have says C(45,5), but I thought it should be C(45,6) or C(40+6-1,6).

Thanks for your help