There are 18 students involved in the class production of arsenic and old lace. a) in how many ways can the teacher cast the play if there are five male roles and seven female roles and the class has nine male and nine female students? b) in how many ways can the teacher cast the play if jean will play the young female part only if jovane plays the male lead? c) in how many ways can the teacher cast the play if all the roles could be played by either a male or a female student?

a. 9Permutation5 × 9Permutation7

= 2743372800

b. for boys there are 9Combination5 = 126 combos

of which jovane

and in 14 of which 1/5 of 70 he will have the lead

for girls there are 9C7 = 36 combos

in 28 of which 7/9 of 36 jean will figure

there are thus a total of 126×36 combos

and we multiplied them by 5!×7!

to get permutations needed for cases:

- jovane chosen & not given lead & jean chosen : 56×28

- jovane not chosen & jean chosen: 56×28

total such combos = 3136

we have now to multiply by 5!×6×6!

to get permutations with restriction,

For instance(5!×7! - 5!×6×6!) = 86400 less for each such combo

so finally, all we have to do

is to subtract 3136×86400 from ans of qa to give

2743372800 - 270950400

= 2472422400

c. 18Permutation12

= 8892185702400

C and D.

Step-by-step explanation:

We are given that Spencer wrote a paragraph for proving that rectangles are parallelograms with congruent diagonals

We have to find the missing step in Spencer's proof

Proof:

According to the given information quadrilateral RECT is a rectangle. By definition of a rectangle , all four angles measure

.Segment ER is parallel to segment CT and segment EC is parallel to segment RT by the converse of the same- side Interior Angles Theorem.

Converse of the same- side angles theorem: When the sum of same side interior angles of a transversal line is then the lines are parallel.

Quadrilateral RECT is then a parallelogram by definition of a parallelogram .Now, construct diagonals ET and CR. Because RECT is a parallelogram ,opposite sides are congruent.Therefore, one can say that segment ER is congruent to segment  CT.Segment TR is congruent to itself by the reflexive Property of equality.The Side-Angle-Side(SAS) Theorem says triangle ERT is congruent to triangle CTR.And because corresponding parts of congruent triangles are congruent (CPCTC),diagonals ET and Cr are congruent.

Answer :Option C and D

Converse of the same-Side Interior angles Theorem