Combination Formulas

Question: 1

Louis Litt wants to form a committee of 5 members out of  3 women and 5 men. In how many ways the committee can be formed if 2 particular ladies are always included in the committee?

Explanation: 

Since there are 3 ladies and 5 gentlemen and a committee of 5 members to be formed from them.

Number of ways such that two ladies are always included in the committee,

^6C_{3}=\frac{6\times5\times4}{6}=20

Question 2:

Jon Snow wants to form a Committee consisting of 5 men and 6 women can be formed out of 8 men and 10 women. In how many ways it is possible?

Solution:

Number ways to select men = ^8C_{5}\times^{10}C_{6}

Number of ways to select women = ^8C_{3}\times^{10}C_{4}

=\left ( \frac{8\times7\times6}{3\times2\times1} \right )\times\left ( \frac{10\times9\times8\times7}{4\times3\times2\times1} \right )

=11760

Question 3:

Robert Ketriz wants to form a team of 6 children for the Student Council Member out of 4 girls and 5 boys such that it contains at least two girls. In how many different ways can the selection be made?

Solution:

Total number of ways in which 6 students can be selected =^9C_6

Selection of atmost 1 girl selection = ^4C_1\times^5C_5

Selection of at least 2 girls = ^9C_6-^4C_1\times^5C_5=84-4=80

Question 4:

Ulrich Nielson’s Family consists of 5 people out of whom only 2 can drive and are to be seated in a five-seater Chevrolet Suburban with 2 seats in front and 3 in the rear. The people who know driving don’t sit together. Only someone who knows driving can sit in the driver’s seat. Find the number of ways all 5 family members can be seated.

Solution:

Number of people who can drive = 2

Number of ways of selecting driver = ^2C_1

Other person who knows driving can be seated only in the rear three seats in 3 ways

Total number of ways of seating the two persons = ^2C_1\times3

Number of ways of seating remaining = 3!

Total number of all five can be seated = ^2C_1\times3\times3! = 36

Question 5:

Accenture’s objective test with all the questions mandatory to be answered can be attempted in 127 ways such that the applicant gets at least one question right. Find the number of ways in which he can answer 4 questions correctly.

Solution:

Any random question can be answered in 2 ways (right or wrong)

Let the number of questions be N

2^N – 1 = 127

Therefore N = 7 

Number of ways in answering 4 answers correctly =^7C_4 = 35