800guy wrote:A bar is creating a new signature drink. There are five possible alcoholic ingredients in the drink: rum, vodka, gin, peach schnapps, or whiskey. There are five possible non-alcoholic ingredients: cranberry juice, orange juice, pineapple juice, limejuice, or lemon juice. If the bar uses two alcoholic ingredients and two non-alcoholic ingredients, how many different drinks are possible?

A. 100
B. 25
C. 50
D. 75
E. 3600

5C2 * 5C2
= (5!/((5-2)! * 2! ) * (5!/((5-2)! * 2! )
= 10 * 10 = 100

Answer is A : 100

Agreed. 5C2*5C2

OA:

The first step in this problem is to calculate the number of ways of selecting two alcoholic and two non-alcoholic ingredients. Since order of arrangement does not matter, this is a combination problem.

The number of combinations of n objects taken r at a time is

C(n,r) = n!/(r!(n-r!))

The number of combinations of alcoholic ingredients is

C(5,2) = 5!/(2!(3!))
C(5,2) = 120/(2(6))
C(5,2) = 10

The number of combinations of non-alcoholic ingredients is

C(5,2) = 5!/(2!(3!))
C(5,2) = 120/(2(6))
C(5,2) = 10

The number of ways these ingredients can be combined into a drink can be determined by the Multiplication Principle. The Multiplication Principle tells us that the number of ways independent events can occur together can be determined by multiplying together the number of possible outcomes for each event.

The number of possible drinks is

= 10 * 10
= 100

this is a combination problem,
and the formula is given as
C(n,r)=n!/ (r!(n-r!))

since we are having two ingredients which are alcoholic and non-alcoholic.
we will find their combination separately,

first let find that of non-alcoholic (na),
using the combination formula, C(n,r)=n!/ (r!(n-r!))
where n=5 and r=2
C(5,2)na= 5!/ (2! (5-2!))
5!=5*4*3*2*1= 120
2!=2*1=2
3!=3*2*1=6
C(5,2)= 120/ (2*6)
=120/12 =10

now, let us find the combination for the alcoholic(a)
C(n,r)a=n!/ (r!(n-r!))
n=5 and r=2
C(5,2)= 5!/ (2! (5-2!))
C(5,2)= 120/ (12)
=10

so we will apply the multiplication rule to determine the number of ways the ingredients can combine together.
C(5,2)na * C(5,2)a = 10*10= 100
so, for this question, the correct option is A.