Multinomial Theorem

Q.

In an examination, the maximum marks for each of three papers are $50$ each.

Maximum marks for the fourth paper are $100$.

The number of ways in which the candidate can score $60\%$ marks in the aggregate is

1. $110556$

2. $110500$

3. $110356$

4. None of these

Q.

$3(x-3)=5(2x+1)$, Then the Value of $x$ Is

Q. Let $f(n,k)$ denote the number of ways in which $k$ identical balls can be coloured with $n$ colours so that there is at least one ball of each colour. Then $f(n,2n)$ must be equal to

1. ${}^{2n}{C}_n$
2. ${}^{2n-1}{C}_{n+1}$
3. ${}^{2n-1}{C}_n$
4. none of these

Q. The number of non-negative integral solutions of the equation $x+y+z+5t=15$ is

1. $196$
2. $224$
3. $312$
4. $364$

Q. In how many ways can $15$ identical blankets be distributed among $6$ persons such that everyone gets atleast one blanket and two particular persons get equal blankets and another three particular persons get equal blankets.

1. $8$
2. $15$
3. $10$
4. $12$

Q. The total number of ways of distributing $21$ books among $5$ children

1. without any restriction is ${}^{25}{C}_4$
2. when each child should get atleast one book is ${}^{16}{C}_4$
3. when each child should get atleast three books is ${}^{10}{C}_4$
4. when each child should get atleast four books is $5$

Q. In how many ways, can we get a sum greater than $17$ by throwing six distinct dice

1. $6^6-({}^{17}{C}_6-6{}^{11}{C}_5)$
2. $6^6-({}^{17}{C}_6+6{}^{11}{C}_5)$
3. $5^6-({}^{17}{C}_6-6{}^{11}{C}_5)$
4. $5^6-({}^{17}{C}_6+6{}^{11}{C}_5)$

Q.

The total number of industries in a particular portion of the country is approximately 1600. If the government has decided to increase the number of industries in the area by 20% every year. Find the approximate number of industries after 2 years

Q. A person buys eight packets of TIDE detergent. Each packet contains one coupon, which bears one of the letters of the word TIDE. If he shows all the letters of the word TIDE, he gets one free packet. If he gets exactly one free packet, then the number of different possible combinations of the coupons is:

1. ${}^7{C}_3$
2. ${}^7{C}_3-1$
3. ${}^8{C}_3$
4. ${}^7{C}_3-2$

Q. A person buys eight packets of TIDE detergent. Each packet contains one coupon, which bears one of the letters of the word TIDE. If he shows all the letters of the word TIDE, he gets one free packet. If he gets exactly one free packet, then the number of different possible combinations of the coupons is:

1. ${}^7{C}_3$
2. ${}^7{C}_3-1$
3. ${}^8{C}_3$
4. ${}^7{C}_3-2$

Q. Let $x+y+z+w=30;$ where $x,y,z,w\in N.$ If $m$ and $n$ denote the number of solutions when no variable may exceed $10$ and each variable is an odd number respectively, then $m+n=$

Q. The number of integral solutions of the equation $x+y+z+t=20,$ such that $x\ge 0,y\ge 1,z\ge 2,t\ge 3,$ is

1. $680$
2. $720$
3. $640$
4. $560$

Q. There are total ${}^{17}{C}_6-k{}^{11}{C}_5$ ways to get a sum of atmost $17$ by throwing six distinct dice , then $k$ is equal to :

1. $4$
2. $5$
3. $6$
4. $8$

Q. Let $n$ and $k$ be positive integers such that $n\ge {}^{k+1}{C}_2$. The number of integral solutions of $x_1+x_2+\cdots +x_k=n,$ $x_1\ge 1,x_2\ge 2,\cdots x_k\ge k$ is

1. $\left({}^{n-{}^k{C}_2}\right)C_k$
2. $\left({}^{n-1-{}^k{C}_2}\right)C_k$
3. $\left({}^{n-1-{}^k{C}_2}\right)C_{k-1}$
4. $\left({}^{n+1-{}^k{C}_2}\right)C_{k-1}$

Q. Let $x+y+z+w=30;$ where $x,y,z,w\in N.$ If $m$ and $n$ denote the number of solutions when no variable may exceed $10$ and each variable is an odd number respectively, then $m+n=$

Q. Number of ways in which $30$ identical things are distributed among six persons is:

1. ${}^{17}{C}_5$ if each person gets odd number of things.
2. ${}^{16}{C}_{11}$ if each person gets odd number of things.
3. ${}^{14}{C}_5$ if each person gets even number of things(excluding $0$).
4. ${}^{15}{C}_{10}$ if each persongets even number of things(excluding $0$).

Q. The number of ways of selecting $10$ books from book store containing unlimited number of Physics, Chemistry, Mathematics and biology books is

Q. The number of ways of selecting $10$ books from book store containing unlimited number of Physics, Chemistry, Mathematics and biology books is

Q. Fifteen identical balls have to be put in five different boxes. Each box can contain any number of balls. The total number of ways of putting the balls into the boxes so that each box contains at least two balls is equal to $k$. Then which of the following is/are divisor of $k$?

1. $9$
2. $7$
3. $15$
4. $21$

Q. Number of positive integral solutions of $15<x_1+x_2+x_3\le 20$ is

Q. The number of non negative integral solutions of the equation $x+y+3z=33$ is

1. $120$
2. $135$
3. $210$
4. $420$

Q. In an examination, the maximum mark for each of the three papers is $50$ and the maximum mark for the fourth paper is $100.$ The number of ways in which the candidate can score $60\%$ marks in aggregate is :

1. $125640$
2. $110551$
3. $97860$
4. None of these

Q.

One or more options can be correct:
Find the coefficient of $x^{10}$ in $(1-x^7)(1-x^8)(1-x^9)(1-x{)}^{-3}$

1. ${}^{12}{C}_2$

2. ${}^{12}{C}_2{-}^5{C}_2{-}^4{C}_2{-}^3{C}_3$

3. $12C_2{-}^5{C}_3{-}^4{C}_2{-}^3{C}_1$

4. ${}^{12}{C}_{10}$

Q. The number of ways of distributing $20$ identical fruits among $5$ people, so that no one receives less than $3$ fruits is

1. $126$
2. $180$
3. $196$
4. $216$

Q.

In how many ways 3 friends Ram, Rajat and Rupesh having 6 one rupee coins, 7 one rupee coins, 8 one rupee coins respectively donate 10 rupee coin collectively?

1. 40

2. 45

3. 47

4. 50

Q. A bag contains $30$ tokens numbered serially from $0$ to $29.$ The number of ways of selecting $3$ tokens from the bag, such that sum of numbers on them is $30$, is

1. $56$
2. $75$
3. $90$
4. $105$

Q. $15$ identical jewels are to be distributed between $P,Q,R$ and $S$. Find the number of ways in which $P$ gets a maximum of $5$ and $Q$ gets at least $2$ jewels.

1. $560$
2. $120$
3. $440$
4. $680$

Q. Number of ways of creating a $5$ digit number with the help of $1,2$ and $3$ in such a way that sum of digits is $8$

1. $32$
2. $28$
3. $36$
4. $30$

Q. The number of integral solutions for the equation $x+y+z+t=20,$ where $x,y,z,t\ge -1,$ is

1. ${}^{20}{C}_4$
2. ${}^{23}{C}_3$
3. ${}^{27}{C}_4$
4. ${}^{27}{C}_3$