Math Is Fun Forum

1) suppose you have 70 books (35 novels, 20 history books and 15 math books). Assume that all 70 books are different
A) in how many different ways can you put 70 books in a row on a shelf?
B) in how many different ways can you choose a set of 12 books to give to a friend?
C) in how many different ways can you choose a set of 4 history books and 8 novels to give to a friend?
D) in how many different ways can you put the 70 books in a row on a shelf if the novels are on the left, the math books are in the middle and the history books are on the right?

2) what is the coefficient of x^40 and y^10 in the expansion of (2x+y)^50?

3) give a simple expression for the value of the following sum (as a function of n):
C(n,0) + 3 * C(n,1) + 3^2 * C(n,2) +...+ 3^n-1 * C(n,n-1) + 3^n * C(n,n).

Your formula should not involve sums or combinatorial symbols like P(n,r) or C(n,r). Using the binomial theorem briefly justify why your answer is correct.

4) you have 20 pennies, 60 nickels and 40 dimes. Assume that the pennies, dimes and nickels are identical. In how many different ways can you put all the coins in a row?

5) assume we are using LaPlace's probability model, where all outcomes are equally likely. An urn contains 70 balls, of which 10 are red, 20 are blue and 40 are green. Let b be a randomly chosen ball.
A) what is p(b is not green)?
B) what is p(b is blue | b is not red)?

Please help with these questions! Thanks!