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

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. 8
2. 20
3. 28
4. 36

Q. The number of ways of selecting one or more letters from the letters AAAA BBB CDE is

1. 158
2. 144
3. 160
4. 159

Q. The number of ways in which a committee of 3 women and 4 men be chosen from 8 women and 7 men. If Mr. X refuses to serve on the committee if Mr Y is a member of the committee is

1. 420
2. 840
3. 1540
4. 1400

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

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

Q.

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

1. 

2. 

3. 

4. 

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. 385
2. 1110
3. 5040
4. 6210

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. How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent ?

1. $8{\times }^6{C}_4{\times }^7{C}_4$
2. $6\times 7{\times }^8{C}_4$
3. $6\times 8{\times }^7{C}_4$
4. $7{\times }^6{C}_4{\times }^8{C}_4$

Q. Consider the letters of the word $MATHEMATICS$.
Possible number of words in which no two vowels are together is

1. $7!{}^8{C}_4\frac{4!}{2!}$
2. $\frac{7!}{2!2!}{}^8{C}_4\frac{4!}{2!}$
3. $\frac{7!}{2!2!2!}{}^8{C}_4\frac{4!}{2!}$
4. $\frac{7!}{2!}{}^8{C}_4\frac{4!}{2!}$

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. $\frac{9!}{2!2!2!}$