Multinomial Theorem

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

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

Q. If $4$ dice are rolled once, the number of ways of getting the sum as $10$ is

1. $64$
2. $84$
3. $126$
4. $80$

Q. The number of non negative integral solution of $x_1+x_2+x_3+4x_4=20$ is

1. $443$
2. $655$
3. $536$
4. none of these

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

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

Q.

If x denotes the number of sixes in four consecutive throws of a dice, then P(x = 4) is

1. 1/1296

2. 4/6

3. 1

4. 1295/1296

Q. If $x_1+x_2+x_3+x_4+x_5=6$, then the difference between the number of non negative integral solutions and the number of positive integral solutions will be

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 number of positive integral solutions of $a·b·c=30$ is$?$

1. $30$

2. $27$

3. $8$

4. $6$

Q.

The coefficient of$x^{103}$ in $(1+x+x^2+x^3+x^4{)}^{199}.(x-1{)}^{201}$ is

1. 8

2. 9

3. 0

4. 11

Q. The number of positive integral solutions of the equation $x_1{x}_2{x}_3{x}_4{x}_5=1050$ is $375n$ when $n\in N$. Then $n=$

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

Q. In how many different ways , can $3$ persons $A,B,C$ having $6$ one rupee coin, $7$ one rupee coin, $8$ one rupee coin, respectively donate $10$ one rupee coin collectively ?

1. $50$
2. $47$
3. $66$
4. $56$

Q. If $\left[x\right]$ be the greatest integer less than or equal to $x,$ then $\sum _{n=8}^{100}\left[\frac{{(-1)}^{n}n}{2}\right]$ is equal to

1. $2$
2. $-2$
3. $0$
4. $4$

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. In how many ways the sum of upper faces of four distinct dice can be six $?$

1. $10$
2. $84$
3. $56$
4. None of these

Q. The number of non negative integral solutions for the equations $a+b+c+d+e=16$ and $a+b+c=7$ is

1. $36$
2. $185$
3. $192$
4. $360$

Q. The total number of positive integral solution of $15<x_1+x_2+x_3\le 20$ is equal to

1. $685$
2. $785$
3. $1125$
4. none of these

Q. In how many ways can $14$ identical toys be distributed among $3$ boys so that each one gets atleast one toy and no two boys get equal number of toys ?

1. $60$
2. $10$
3. $20$
4. none of these

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

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

Q.

The data in the following table show that the percentage of adults in the United States who are currently married is declining.

$$\begin{array}{|cc|}\hline \text{Year}& \text{Percent of Adults}\\ \text{Who Are Married}\\ 1960& 72.2\mathrm{\%}\\ 1980& 62.3\\ 2000& 57.4\\ 2010& 51.4\\ 2012& 50.5\\ \hline\end{array}$$
Assuming that the percentage of adults who are married will continue to decrease according to the exponential decay model:
a) Use the data for $1960and2012$ to find the value of $k$ and to write an exponential function that describes the percent of adults married after time $t$, in years, where $t$ is the number of years after $1960$ .
b) Estimate the percent of adults who are married in $2015\text{and in}2018$.
c) At this decay rate, in which year will the percent of adults who are married be $40\mathrm{\%}?$

Q. For the equation $x_1+x_2+x_3\le 13$, the correct option(s) is/are

1. The number of non negative integral solution will be $560$
2. The number of positive integral solution will be $220$
3. The number of non negative integral solution, when $x_2\ge 3$ will be $286$
4. The number of non negative integral solution, when $x_1\ge 2,x_2\ge 4$ will be $120$

Q. Number of ways in which $25$ identical things can be distributed among five persons if each gets odd number of things is

1. ${}^{25}{C}_4$
2. ${}^{12}{C}_8$
3. ${}^{13}{C}_3$
4. ${}^{14}{C}_4$

Q. The number of integral solutions of $x+y+z=0$ with $x\ge -5,y\ge -5,z\ge -5$ is

1. $134$
2. $138$
3. $136$
4. $140$

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

Q.

How do you find $2A-3B$ given $A=\left[\begin{array}{cccc}5& -2& 3& 1\end{array}\right]$ and $B=\left[\begin{array}{cccc}-2& 3& 1& 0\end{array}\right]$ ?

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. $560$