can anyone walk me through this step?

Question

a) find the coefficient of x^3y^5 in the expansion of (x+y)^8

b)Pigeonhole Principle/Multiplication Principle/Addition Principle
If you have brown, blue, black, green, tan, and white socks, in a drawer, how many socks must you remove from the drawer in order to be sure of having a match?

c) how many numbers must you choose from the set {1,2,3,....,15} to be sure that you have at least one even number? at least one odd number?

Answer 1

a) From the binomial theorem, the expansion of (x+y)^8 will have terms of the form [8!/(a!*(8-a)!]*x^a*y^(8-a). The term in question is a=3; thus, [8!/(3!*5!)]*x^3*y^5. The coefficient simplifies to 56. (See the link to the binomial theorem below.)

b) The only way to ensure that you have a matching pair of socks is to pull out 7 socks. On the first six pulls, you might end up with one each of the 6 different colors. Not likely, but not a 0% likelihood either. The only way to guarantee it is to pull 7 socks. (See the link to the birthday paradox below... the only way to guarantee that two people will share a birthday is to have 366 people, despite intuition telling us otherwise!)

c) Similarly, you have 8 odd numbers in the set, the rest being even. Thus, pulling one such number out of a hat, you must pull 9 numbers to guarantee that one is even. With 7 even numbers in the set, you must pull 8 numbers to guarantee that one is odd.

Answer 2

b) No more than 7, since you only have 6 colors. The 7th color HAS to match one of the 6, even if you draw out all different colors first.(It could match in as little as 2 tries.)

c) You could take as many as 9 tries to get an even number, since there are 8 odd numbers in the set, and as many as 8 tries for an odd number with 7 even numbers in the set.

Answer 3

the nth term looks like 8C(n-1) x^(9-n) y^(n-1). since the x exponent is 3, must be 6th term. check: since y exponent is 5, must be 6th term. so coefficient is 8C5 = 8C3 = (8•7•6)/(1•2•3) =56

after you've drawn 1 sock of each color (worst case), which is 6, the 7th sock must match one of them.

there are 7 even numbers, 8 odd. to be sure of at least 1 even, first draw all the odd, so need 9 draws. to be sure of 1 odd, first draw all the even, so need 8 draws. this assumes drawing without replacement.

Answer 4

a) 56
(x + y)² = (x² + 2xy + y²)
(x + y)³ = (x³ + 3x²y + 3xy² + y³)
...
(x + y)^8 = (x^8 + 8x^7y + 28x^6y^2 + 56x^5y^3 + 70x^4y^4 + 56x^3y^5 + 28x^2y^6 + 8xy^7 + y^8)

b) 7. Since you have six different colors, you may pull out a sock of each different color on your first six tries. The seventh will definitely get you a matching pair.

c) 9 (for even), 8 (for odd). By the same reasoning as question b).