Number of ways of Arranging "n" things taken "k" at a time, With and Without Repetition

Q.

How many $3$ letter words with or without meaning can be formed out of the letters of the word $SMOKE$ when repetition of words is allowed?

Q. A license plate begins with three letters. If the possible letters are $A,B,C,D,E,F,G$ and $H,$ how many different permutations of these letters can be made if no letter is used more than once?

1. $336$
2. $8^3$
3. $3!$
4. $8!$

Q. Find the number of $5$ letter words, with or without meaning, which can be formed out of the letters of the word $AUGUST,$ where the repetition of the letters is not allowed.

1. $3125$
2. $120$
3. $720$
4. $50$

Q.

If it is possible to make a meaningful word with the first, the fifth, the sixth and the eleventh letters of the word $COURAGEOUSLY$, which of the following will be third letter of that word?

If no such word can be made, give $X$ as the answer and if more than one such word can be made, give $M$ as the answer

1. $O$

2. $A$

3. $G$

4. $X$

Q. From among $30$ teachers in a school, one principal and one vice principal are to be appointed. In how many ways can this be done?

1. $780$
2. $8700$
3. $870$
4. $660$

Q. A zip code contains 3 digits. How many different zip codes can be made with the digits 0–9 if no digit is used more than once and the first digit is not 0?

Q. In how many ways can 7 students stand in a queue?

1. $5040$
2. $7!$
3. $6!$
4. (a) and (b) above.

Q. The number of permutations of $n$ objects taken $r$ at a time is determined by the following formula:

1. $\frac{r!}{(r-n)!}$
2. $\frac{n!}{r!(n-r)!}$
3. $\frac{r!}{(n-r)!}$
4. $\frac{n!}{(n-r)!}$

Q.

How many 3-digit numbers can be formed from the digits 5, 6, 7, 8 and 9 assuming that

(i) Repetition of the digits is allowed?

(ii) Repetition of the digits is not allowed?

1. 27 and 6
2. 15 and 12
3. 125 and 60
4. 60 and 15

Q.

What is an example of a promissory note?

Q. How many numbers between 100 and 1000 can be formed without repetition using the digits from 1 to 9?

1. $\frac{9!}{3!}$
2. $\frac{9!}{(9-3)!}$
3. $3!$
4. None of the above

Q. ${}_5^{8}P$ means

1. permutation of 5 objects taken 8 at a time
2. permutation of 3 objects taken 8 at a time
3. permutation of 8 objects taken 3 at a time
4. permutation of 8 objects taken 5 at a time

Q. How many distinct four-letter words can be formed using the word $\text{CENTURY}$.

1. $840$
2. $720$
3. $5040$
4. $120$

Q. Which of the following is a correct representation for permutation of $10$ objects taken $6$ at a time

1. ${}_{10}{P}_6$
2. ${}_{10}^{6}P$
3. ${}^6{P}_{10}$
4. ${}_{10}{P}^4$

Q. Four books, one English, one science, one social science, and one mathematics, are to be arranged in the shelf. In how many ways this can be done?

1. $6$
2. $4!$
3. $5!$
4. $36$

Q. Given 3 flags of different colours, how many different signals can be generated, if a signal requires the use of 2 flags one below the other?