Question

a) When the expression (A+ a)(B + b)(C + c)(D + d)(E + e) is multiplied out, how many terms will have three uppercase letters?

b) How many ways are there to pick a combination of k things from {1, 2,...,n} if the elements 1 and 2 cannot both be picked? d) 2 How many ways are there to put eight rooks on a chessboard so that no one rook can capture another? (This means that no two are in the same row or same column.) How many ways are there to put five noncapturing rooks on a chessboard?

c) We are given n coins, where n ≥ 2. All but one of the coins are the same weight and the other is heavier. We have a balance scale. Assertion: One weighing suffices to discover
whichcoinisheavier. “Proof”: By induction. When n =2 the result
is clear. Suppose we have proved the result for k coins. We are now given k+1 coins. We proceed as follows. Set one coin aside. Apply the procedure for k coins to the remaining k coins. If we find the heavy coin then we are finished. If not, then the heavy coin is the one we set aside. Thus we have a procedure for k +1 coins. This can’t be right. Where’s the error?

Answer 1