Use an organized list, a tree diagram, or the fundamental counting principle to find the number of permutations. the number of ways 6 shirts can be arranged in a closet

Find integers x and y so that 100x+57y=gcd(100,57).

Determine the end behavior of the graph of each polynomial function. y = 6x + 25 + 4x4 - x²

Predict the sign of $\Delta S ^ { \circ }$ for $\mathrm { H } 2 \mathrm { S } ( \mathrm { g } ) + 1 / 8 \mathrm { O } 8 ( \mathrm { s } ) \rightarrow \mathrm { i } \mathrm { Sg } ( \mathrm { j } ) + \mathrm { H } 20 ( \mathrm { g } )$

If $a , b$ are integers and 3$\left\langle a$ or $3 / b ,$ show that 3$/ \left( a ^ { 2 } + b ^ { 2 } \right)\right.$

Licensed therapists are mandated by law to report child abuse by their clients. This law requires the therapist to breach confidentiality and possibly lose the client’s trust. A national survey of licensed psychotherapists was conducted to investigate clients’ reactions to legally mandated child-abuse reports ( American Journal of Orthopsychiatry , Jan. 1997). The sample consisted of 303 therapists who had filed a child-abuse report against at least one of their clients. The researchers were interested in finding the best predictors of a client’s reaction ( y ) to the report, where y is measured on a 30-point scale. (The higher the value, the more favorable was the client’s response to the report.) The independent variables found to have the most predictive power are as follows:

$$x_1:$$

Therapist's age (years)

$$x_2:$$

Therapist's gender (1 if male, 0 if female)

$$x_3:$$

Degree of therapist's role strain (25 - point scale)

$$x_4:$$

Strength of client - therapist relationship (40 - point scale

$$x_5:$$

Type of case (1 if family, 0 if not)

$$x _ { 1 } x _ { 2 } : \text { Age } \times \text { Gender interaction }$$

a. Hypothesize a first-order model relating y to each of the five independent variables and the interaction term. b. Give the null hypothesis for testing the contribution of

$$x_4,$$

strength of client–therapist relationship, to the model. c. The test statistic for the test suggested in part b was t = 4.408, with an associated p -value of .001. Interpret this result. d. The estimated

$$\beta$$

coefficient for the

$$x_1 x_2$$

interaction term was positive and highly significant (p < .001). According to the researchers, “This interaction suggests that c as the age of the therapist increased, c male therapists were less likely to get negative client reactions than were female therapists.” Do you agree? e. For the model presented here,

$$R^2 = .2946$$

. Interpret this value.

There are a total of 968 students enrolled in the three grades at Middletown Middle School. There are 329 students in the sixth grade and 328 students in the seventh grade. Write and solve an equation to find how many students are in the eighth grade.

Detennine the fixed field of complex conjugation on $\mathbb{C}$.

Solve each inequality and graph the solution set on a number line.

$$x - 3 > 2$$

Factor the following positive integers into primes. a. 25. b. 4200. c. $10^{10}$. d. 19. e. 1.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

$$_ { 7 } P _ { 3 } = 3 ! _ { 7 } C _ { 3 }$$

Solve.

$$20 x + 50 = 250$$

You have a balance of $213 in a savings account. You withdraw$68 one day and $128 the next day without making any deposits. What is your new balance?

The particles of a substance move _______ as the temperature of the substance increases.