Find the error in the following reasoning: “Consider forming a poker hand with two pairs as a five-step process. Step 1: Choose the denomination of one of the pairs. Step 2: Choose the two cards of that denomination. Step 3: Choose the denomination of the other of the pairs. Step 4: Choose the two cards of that second denomination. Step 5: Choose the fifth card from the remaining denominations. There are

$$\left( \begin{array} { c } { 13 } \\ { 1 } \end{array} \right)$$

ways to perform step 1,

$$\left( \begin{array} { l } { 4 } \\ { 2 } \end{array} \right)$$

ways to perform step 2,

$$\left( \begin{array} { c } { 12 } \\ { 1 } \end{array} \right)$$

ways to perform step 3,

$$\left( \begin{array} { l } { 4 } \\ { 2 } \end{array} \right)$$

ways to perform step 4, and

$$\left( \begin{array} { c } { 44 } \\ { 1 } \end{array} \right)$$

ways to perform step 5. Therefore, the total number of five-card poker hands with two pairs is

$$13 \cdot 6 \cdot 12 \cdot 6 \cdot 44 = 247,104$$

."

In Morse code, symbols are represented by variable-length sequences of dots and dashes. (For example,

$$A = \cdot -, 1 = \cdot - - - - , ? = \cdot \cdot - - \cdot \cdot )$$

. How many different symbols can be represented by sequences of seven or fewer dots and dashes?

Let $T:R^n\to R^m$ be a linear transformation, and $\mathcal{B}=\{b_1,b_2,\dots,b_n\}$ and $\mathcal{C}=\{c_1,c_2,\dots,c_m\}$ be bases for $R^n$ and $R^m$, respectively. Let $B$ and $C$ be the matrices whose columns are the vectors in $\mathcal{B}$ and $\mathcal{C}$, respectively. Prove the following results: (a) If $A$ is the standard matrix $T$, then $[T]^C_B=C^{-1}AB$. (b) If $U:R^n\to R^m$ is linear and $s$ is any scalar, the (i) $[T+U]_B^C=[T]_B^C+[U]_B^C$; (ii) $[sT]_B^C=s[T]_B^C$; (iii) $[T(v)]_C=[T]_B^C[v]_B$, for any vector $v$ in $R^n$.

How many pairs of two distinct integers chosen from the set {1, 2, 3, ..., 101} have a sum that is even?

find the slope of the line with inclination θ. θ = π/6 radian

Verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. $\frac{\tan x+\cot y}{\tan x \cot y}=\tan y+\cot x$

Evaluate the integral.

$$\int \frac { x ^ { 5 } + 3 x ^ { 4 } } { x ^ { 2 } } d x$$

Multiply.

$$12(a+3)$$

Find the coefficient of the given term when the expression is expanded by the binomial theorem.

$$p ^ { 16 } q ^ { 7 }$$

in

$$\left( 3 p ^ { 2 } - 2 q \right) ^ { 15 }$$

What aldehydes are formed when the following amino acids are treated with ninhydrin? a. tyrosine, b. leucine, c. arginine

Factor each expression. $4 x^{2}-4 x+1$

Is disk scheduling, other than FCFS scheduling, useful in a single-user environment? Explain your answer.

Suppose that three computer boards in a production run of forty are defective. A sample of five is to be selected to be checked for defects. a. How many different samples can be chosen? b. How many samples will contain at least one defective board? c. What is the probability that a randomly chosen sample of five contains at least one defective board?

Find the exact distance between the pair of points. (-2, -1), (-5, 8)