Answer:
Explanation:
Step1
Definition permutation (order is important):
P(n,r)= n!/(n−r)!
Definition combination (order is not important):
C(n,r) =( n/r )= n!/r!(n−r)!
Step2
a) The order of the students does not matter, because a different order results in the same students being selected, and thus we should use the definition of combination.
We are selecting 3 of the 17 students in the class (of which 3 are math majors and 14 are not math majors).
# of possible outcomes=(17,3)=17!/3!(17−3)!=17!/3!14!=680
There areC(14,3) ways to select 3 of the 14 no math majors.
# of favorable outcomes=C(14,3)= 14!/3!(14−3)! = 14!/3!11! =364
The probability is the number of favorable outcomes divided by the number of possible outcome.
P(no math majors)= # of favorable outcomes/# of possible outcomes
= 364/680= 91/170
≈0.5353=53.53%
Step 3
(b) By part (a):
P(no math majors)=91/170≈0.5353=53.53%
Use the Complement rule
Complement rule:
P(E’)=1−P(E)
P(at least one math major)=1−P(no math majors)=1−91/170
=79/170≈0.4647=46.47%
Step 4
(c) The order of the students does not matter, because a different order results in the same students being selected, and thus we should use the definition of combination.
We are selecting 3 of the 17 students in the class (of which 3 are math majors and 14 are not math majors).
# of possible outcomes=C(17,3)= 17!/3!(17−3)!=17! /3!14!=680
There areC(14,1) ways to select 1 of the 14 no math majors and there are
C(3,2) ways to select 2 of the 3 major majors.
# of favorable outcomes=C(14,1)⋅C(3,2)=14⋅3=42
P(2 math majors)= # of favorable outcomes/# of possible outcomes
= 42/680= 21/340=approx0.0618=6.18%