91Ó°ÊÓ

How many permutations are there of \(a, b, c, d ?\)

Expand \((x+y)^{4}\) using the Binomial Theorem.

Use the Multiplication Principle. How many dinners at Kay's Quick Lunch (Figure 6.1 .1 ) consist of one appetizer and one beverage? \(2 \cdot 4\)

Prove that among a group of six students, at least two received the same grade on the final exam. (The grades assigned were chosen from \(A, B, C, D, F .)\)

Expand \((2 c-3 d)^{5}\) using the Binomial Theorem.

Suppose that a coin is flipped and a die is rolled. List the members of the event "the coin shows a head and the die shows a number less than 4 ".

Use the Multiplication Principle. The Braille system of representing characters was developed early in the nineteenth century by Louis Braille. The characters, used by the blind, consist of raised dots. The positions for the dots are selected from two vertical columns of three dots each. At least one raised dot must be present. How many distinct Braille characters are possible?

How many permutations are there of 11 distinct objects?

How many 5-permutations are there of 11 distinct objects?

Suppose that six distinct integers are selected from the set \(\\{1,2,3,4,5,6,7,8,9,10\\} .\) Prove that at least two of the six have a sum equal to 11. Hint: Consider the partition {1,10} , {2,9},{3,8},{4,7},{5,6}.