$2\mathrm{.}1$ Given n bits, how many distinct combinations of the n bits exist?

$2\mathrm{.}2$ There are $26$ characters in the alphabet we use for writing English. What is the least number of bits needed to give each character a unique bit pattern? How many bits would we need to distinguish between upper$-$and lowercase versions of all $26$ characters?

$2\mathrm{.}3$ a$.$ Assume that there are about $400$ students in your class. If every student is to be assigned a unique bit pattern, what is the minimum number of bits required to do this?

b$.$ How many more students can be admitted to the class without requiring additional bits for each student$$s unique bit pattern?

$2\mathrm{.}4$ Given n bits, how many unsigned integers can be represented with the n bits? What is the range of these integers?

$2\mathrm{.}5$ Using five bits to represent each number, write the representations of $7$ and $7$ in $1$s complement, signed magnitude, and $2$s complement integers.

$2\mathrm{.}6$ Write the six$-$bit $2$s complement representation of $32.$

$2\mathrm{.}7$ Create a table showing the decimal values of all four$-$bit $2$s complement numbers.

$2\mathrm{.}8$ What is the largest positive number one can represent in an eight$-$bit $2$s complement code? Write your result in binary and decimal.

What is the greatest magnitude negative number one can represent in an eight$-$bit $2$s complement code? Write your result in binary and decimal.

What is the largest positive number one can represent in n$-$bit $2$s complement code?

What is the greatest magnitude negative number one can represent in n$-$bit $2$s complement code?

$2\mathrm{.}9$ How many bits are needed to represent Avogadro$$s number $(6\mathrm{.}021023)$ in $2$s complement binary representation?

$2\mathrm{.}10$ Convert the following $2$s complement binary numbers to decimal.

a$.1010$

b$.01011010$

c$.11111110$

$2\mathrm{.}11$ Convert these decimal numbers to eight$-$bit $2$s complement binary numbers.

a$.102$

b$.64$

c$.33$

d$.128$

e$.127$