Number of Ways of Selecting "k" Things Out of "n" Things at a Time

Q.

Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from $10$ white, $9$ green and $7$ black balls is?

1. $880$

2. $629$

3. $630$

4. $879$

Q. In how many ways can a student choose 6 courses out of 10 courses?

1. $5040$
2. $210$
3. $6!$
4. $\frac{6!}{3!}$

Q. How many different selections of 6 books can be made from 10 different books?

1. ${}^6{C}_{10}$
2. $\frac{7!}{4!}$
3. $5040$
4. (a) and (b) above.

Q. Number of ways of selecting ${}^{\prime }{n}^{\prime }$ objects taken ${}^{\prime }{r}^{\prime }$ at a time is given as ${}^n{C}_r=\frac{n!}{(n-1)!}$, where order does not matter.

1. True
2. False

Q. From a class of 15 boys and 12 girls, 9 students are to be chosen for the competition. In how many ways can the selection be made?

1. ${}^{15}{C}_9$
2. ${}^{}12C_9$
3. ${}^{27}{C}_9$
4. All of the above.

Q. How many different combinations of product can be obtained by multiplying three numbers from $12,13,14,15,16,19$ without using same number twice?

1. $20$
2. $\frac{6!}{3!}$
3. $6!$
4. $120$

Q. Adrienne has free passes to a music concert and can take $2$ friends along, but she has $5$ friends who could go with her. In how many ways can she select $2$ friends out of $5$?

1. $10$
2. $20$
3. $30$
4. $15$

Q. Number of ways of choosing ${}^{\prime }{n}^{\prime }$ things taken ${}^{\prime }{r}^{\prime }$ at a time is given as :

Statement 1: Number of ways if Order matters $=\frac{n!}{(n-r)!}$
Statement 2: Number of ways if Order does not matter$=\frac{n!}{(n-r)!\times r!}$

Select the correct statement

1. Statement $1$ is true and statement $2$ is false
2. Statement $1$ is false and statement $2$ is true
3. Both the statements are true
4. Neither of the statements are true

Q. A team is to be formed consisting of $2$ players for a tennis tournament. $3$ players P, A, and R are available for selection. In how many ways can the team be selected?

1. $5$
2. $4$
3. $3$
4. $6$

Q. Select the correct notations for combination

1. ${}^n{C}_r$
2. $\left({}_r^n\right)$
3. $C{}_k^n$
4. ${}_r{C}^n$
5. $\left(nr\right)$
6. ${}^{n}C{}^k$
7. ${}_n{C}_r$
8. $n{C}_r$
9. $C{}_r^n$

Q. For the cricket world cup, playing 11 is to be selected out of 16 players in the squad. In how many ways can this be done?

1. $\frac{16!}{5!\times 11!}$
2. $\frac{16!}{11!}$
3. $\frac{16!}{5!}$
4. None of the above

Q. A cow has $8$ different pairs of roller blades. In how many ways can she select $2$ pairs to go skating with her friends?

1. $24$
2. $28$
3. $32$
4. $36$

Q. A bag contains $7$ balls of same color. What is the correct expression to find the number of ways for selecting $5$ balls?

1. ${}^7{\text{P}}_5$
2. ${}^7{\text{C}}_2$
3. ${}^7{\text{C}}_3$
4. ${}^7{\text{P}}_2$

Q. The value of $k$ is if ${}^6{\text{C}}_k=\frac{6!}{\left(4\right)!\times k!}$

1. $5$
2. $2$
3. $4$
4. $3$