Acountry has two political parties, the demonstrators and the repudiators. suppose that the national senate consists of 100 members, 44 of which are demonstrators and 56 of which are repudiators.
(a) how many ways are there to select a committee of 10 senate members with the same number of demonstrators and repudiators?
(b)suppose that each party must select a speaker and a vice speaker. how many ways are there for the two speakers and two vice speakers to be selected?

(a) 4.148 x 10^(12) ways

(b) 5,827,360 ways

Step-by-step explanation:

Number of Demonstrators (D)  = 44

Number of Repudiators (R) = 56

(a)

5 senate members must be Repudiators and 5 must be demonstrators, assuming that the order at which they are selected is irrelevant:

(b)

Since there are two different positions, (speaker and vice speaker), order is important in this situation, and the total number of ways to select two senators from each party is:

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.