91Ó°ÊÓ

How many permutations are there for the letters of the name Bathsheba? Solomon? Ahab? your own name?

The English alphabet contains 26 letters, including five vowels. In each case determine how many words of length five are possible provided that: (a) Words contain at most two distinct vowels (b) Words contain at most one letter that is a vowel (c) Words contain at least four distinct vowels

How many 7 -digit numbers are there such that the digits are distinct integers taken from \((1,2, \ldots, 9)\) and the integers 5 and 6 do not appear together in either order?

A convex polygon is a polygon such that any line segment joining two points inside the polygon lies entirely inside the polygon. If no 3 of the 15 diagonals of a convex, sixsided polygon intersect at a point common to all three, into how many line segments are the diagonals divided by their intersection points? Can you conjecture and prove a general result for an \(n\) -sided convex polygon?

A bridge hand consists of 13 cards dealt from the 52 -card deck. Bridge involves four players named North, East, South, and West. How many ways can the cards be dealt so that the game can be played?

There are six points in a plane, no three of which are collinear, In how many ways can you draw a pair of triangles with the six points as vertices.

Find the number of paths from \(A\) to \(F\) in the following diagram with six letters. A path can only go through letters that are consecutive, either horizontally or vertically, and it goes only to the right or up at each step. $$ \begin{array}{lllll} \text { F } & & & & \\ \text { E } & \text { F } & & & & \\ \text { D } & \text { E } & \text { F } & & & \\ \text { C } & \text { D } & \text { E } & \text { F } & & \\ \text { B } & \text { C } & \text { D } & \text { E } & \text { F } \\ \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { P } \end{array} $$ Prove that a similar path with \(n\) letters has \(2^{n-1}\) paths from the lower left corner to any letter in the rightmost position in a row.

Internet Addresses: IPv4 and IPv6. The Internet requires an address for each machine that is connected to it. The address space of the addressing architecture of Internet Protocol version 4 (IPv4) consists of a 32 -bit field. Since not every combination of bits can be used as an address, plans are underway to change the address space to a 128 -bit field in IPv6. The 32 -bit IPv4 addresses are usually written in a form called dotted decimal. The 32 bit address is broken up into four 8 -bit bytes, and these bytes are then converted to their equivalent decimal form and separated by dots. For example. $$ \begin{array}{ll} 1000000000000011 & 00000010000000011 \end{array} $$ is written as 128.3 .2 .3 , which is obviously more readable. The 128 -bit IPv6 addresses are divided into eight 16 -bit pieces. Each 16 -bit piece is converted to its equivalent hexadecimal value (each sequence of 4 bits is converted to one hexadecimal digit). The eight four-character hexadecimal strings are separated by colons. It is not prac. tical to list 128 bits and show the conversion to the final IPv6 address form. As an example of what you might end up with, however, we show one IPv6 address: \(\mathrm{FFDC} \cdot \mathrm{BA} 98: 7654 \cdot 3210: \mathrm{FEDC}: \mathrm{BA} 98: 7654 \cdot 3210\) How many IPv4 addresses are possible?

Construct the first 10 rows of Pascal's triangle.

Two committees of five persons each must be chosen from a group of 375 people. If the committees must be disjoint, in how many ways can the committees be chosen? If the committees need not be disjoint, in how many ways can this be done?