Combinations practice · Mixed difficulty

The value of $\sum_{k=0}^{6}{ }^{51-k} C_{3}$ is

Seven people leave their bags outside temple and while returning after worshiping the deity, picked one bag each at random. In how many ways at least one and at most three of them get their correct bags?

If the total number of $m$ elements subsets of the set $A = \left\{ {{a_1},{a_2},{a_3}, \ldots {a_n}} \right\}$ is $\lambda$ times the number of 3 elements subsets containing ${a_4}$, then $n$ is

The sum $\sum\limits_{0 \leqslant i} {\sum\limits_{ \leqslant j \leqslant 10} {\left( {^{10}{C_j}} \right)\left( {^j{C_i}} \right)} }$ is equal to

The number of ways in which a mixed double game can be arranged from amongst 9 married couples if no husband and wife play in the same game is

The number of non-negative integral solutions to the system of equations $x+y+z+u+t=20$ and $x+y+z=5$ is-

The value of $\left( {{}^{21}{C_1} - {}^{10}{C_1}} \right) + \left( {{}^{21}{C_2} - {}^{10}{C_2}} \right) + \left( {{}^{21}{C_3} - {}^{10}{C_3}} \right) + \left( {{}^{21}{C_4} - {}^{10}{C_4}} \right) + ... + \left( {{}^{21}{C_{10}} - {}^{10}{C_{10}}} \right)$ is

If $n$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to:

In a network of railways, a small island has 15 stations. The number of different types of tickets to be printed for each class, if every station must have tickets for other station, are

There are two urns. Urn $A$ has 3 distinct red balls and urn $B$ has 9 distinct blue balls. From each urn, two balls are taken out at random and then transferred to the other. The number of ways in which this can be done, is

A man $X$ has 7 friends, 4 of them are ladies and 3 are men. His wife $Y$ also has 7 friends, 3 of them are ladies and 4 are men. Assume and $Y$ have no common friends. Then, the total number of ways in which $X$ and $Y$ together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of $X$ and $Y$ are in this party, is

If $n \in N$, then value of the expression ${ }^{n-2} C_r+2 .^{n-2} C_{r-1}+{ }^{n-2} C_{r-2}=$

Number of ways in which 6 different toys can be distributed among two brothers in $1: 2$, is

If ${\;^8}{C_r} - {\;^7}{C_3} = {\;^7}{C_2}$, then $r$ is equal to

Let $A$ be the set of 4 -digit numbers $a_{1} a_{2} a_{3} a_{4}$ where $a_{1} > a_{2} > a_{3} >$ $a_{4}$. Then $n(A)$ is equal to

5-digit numbers are formed using 2, 3, 5, 7, 9 without repeating the digits. If $p$ is the number of such numbers that exceeds 20000 and $q$ be the number of those that lie between 30000 and 90000 then $p : q$ is:

A binary sequence is an array of 0's and 1's. The number of $n -$ digit binary sequence which contain even number of 0's is

The number of diagonals in a polygon of $m$ sides is

If in a chess tournament each contestant plays once against each of the other and in all 45 games are played, then the number of participants is

Number of positive unequal integral solutions of the equation $x+y+z=12$ is

If $a,\;b,\;c \in N,$ The anumber of points having position vector $a\mathbf{\hat i} + b\mathbf{\hat j} + c\mathbf{\hat k}$ such that $6 \le a + b + c \le 10$, is

The number of positive integers satisfying the inequality ${ }^{n+1} C_{n-2}-{ }^{n+1} C_{n-1} \leq 100$ is

The maximum of points in to which 4 circles and 4 straight lines intersect is

A gentlemen hosts a party of $(m+n)$ guests and places ' $m$ ' at one round table and the remaining ' $n$ ' at the other round table. Number of ways the guests can be arranged is

The greatest possible number of points of intersection of 8 straight lines and 4 circle is

The number of ways in which 20 letters $x_1, x_2, … x_{10}, y_1, y_2, …, y_{10}$ be arranged in a line so that suffixes of the letters $x$ and also those of $y$ are respectively in ascending order of magnitude is

${\;^n}{C_r} + 2{\;^n}{C_{r - 1}} + {\;^n}{C_{r - 2}}$ is equal to

The number of positive odd divisors of 216 is

If ${x}_{{k}} \geq 0, {k}=1,2,3,4,5$ and ${x}_1+{x}_2+{x}_3+{x}_4+{x}_5=20$ and ${x}_1+{x}_2+{x}_3=5$ then the number of integral solution of $\left(x_1, x_2, x_3, x_4, x_5\right)$

If ${}^n{C_4},{}^n{C_5},\mathrm{ }\,\,\mathrm{and }\,\,{}^n{C_6}$ are in AP, then n can be