Combination of r Things from n Things When All Are Not Different

Q.

At an election, a voter may vote for any number of candidates, not greater than the number to be elected.There are 10 candidates and 4 are to be elected. If the voter votes for atleast one candidate, then the number of ways in which he can vote is

1. 5040

2. 6210

3. 385

4. 1110

Q. The total number of $6$ digit numbers that can be made using digits $1,2,3,4$, if all the digits should appear in the number at least once is

1. $1560$
2. $840$
3. $1080$
4. $480$

Q. Number of ways of selecting 6 shoes, out of 8 pairs of shoes, having exactly two pairs is

1. 1680
2. 240
3. 120
4. 3360

Q. Consider the letters of the word $MATHEMATICS$.
Possible number of words taking all letters at a time such that in each word both $M$’s are together and both $T$’s are together but both $A$’s are not together is

1. $\frac{11!}{2!2!2!}-\frac{10!}{2!2!}$
2. $7!\times {}^8{C}_2$.
3. $\frac{6!4!}{2!2!}$
4. $\frac{9!}{2!2!2!}$

Q. Messages are conveyed by arranging $4$ white, $1$ blue and $3$ red flags on a pole. Flags of the same colour are alike. If a message is transmitted by the order in which the colours are arranged, the total number of messages that can be transmitted if exactly six flags are used is

Q. Number of words that can be formed by taking $4$ letters at a time out of the letters of the word $MATHEMATICS$ is

1. $1680$
2. $1698$
3. $2436$
4. $2454$

Q. In a shop there are five types of ice creams available. A child buys six ice creams.
Statement 1: The number of different ways the child can buy the six ice-creams is
${}^{10}{C}_5$.
Statement 2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A's and 4 B's in a row.

1. Statement 1 is false, statement 2 is true.
2. Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.
3. Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.
4. Statement 1 is true, statement 2 is false.

Q. The number of $4$ letter words (with or without meaning) that can be made from the eleven letters of the word "EXAMINATION" is

Q. Number of ways of selecting none or more of $10$ identical things is

1. $2$
2. $10$
3. $11$
4. $9$

Q. The solution set of $\frac{x^2-7x+10}{14x-x^2-45}\ge 0$ is

1. $[2,9]$
2. $[2,9)$
3. $[2,5)\cup (5,9)$
4. $[2,5)\cup (5,9]$

Q. There are $2$ women participating in a chess tournament. Every participant played $2$ games with the other participants. The number of games that the men played between themselves exceeded by $66$ as compared to the number of games that the men played with the women. If the number of participants is $k$ then the value of $k-6$ is

Q. Number of words that can be formed by using the $4$ letters of the word $MISSISSIPPI$ is

1. $192$
2. $148$
3. $176$
4. $164$

Q.

The number of arrangements of the letters of the word ‘NAVA NAVA LAVANYAM’ which begin with N and end with M is :

1. $\frac{\angle 16}{\angle (\angle 3{)}^2}$

2. $\frac{\angle 16}{\angle 7\angle 3}$

3. $\frac{\angle 14}{\angle 7\angle 3\angle 2}$

4. $\frac{\angle 14}{\angle 7\angle 3}$

Q. Number of ways of selecting none or more of $10$ identical things is

1. $2$
2. $10$
3. $11$
4. $9$

Q. Consider the word ${}^{\prime \prime }SUCCES{S}^{\prime \prime }$, then the total number of possible words

1. when all the letters are taken at a time is $420$
2. when All $S^{\prime }s$ are not together is $360$
3. when no two $S^{\prime }s$ are together is $180$
4. taking $4$ letters at a time is $92$

Q. $\text{Let us consider the word 'MOTIHARI'. Six letters are choosen from the given word and}\text{arranged in all possible ways to form a new word (may be meaningful or non-meaningful),}\text{then the number of new words so formed are}$

1. $10440$
2. $11440$
3. $9440$
4. $12440$

Q.

Let $A=\{lettersofNEWJERSEY\}$and $B=\{lettersofNEWYORK\}$

Find $A\cup B$

Q. In a polygon no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon be 70 then the number of diagonals of the polygon is

1. 20
2. 28
3. 8
4. 36