Combinations practice · Mixed difficulty

In a football championship, there were played 153 matches. Every team played one match with each other. The number of teams participating in the championship is

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets atleast one ball, is

The number of five digit numbers that contain 7 exactly once is

$A$ is a set containing $n$ elements. A subset $P$ of $A$ is chosen. The set $A$ is reconstructed by replacing the elements of $P$. A subset $Q$ of $A$ is again chosen, the number of ways of choosing so that $(P \cup Q)$ is a proper subset of $A$, is

If $(1+x)^n=\sum\limits_{{r=0}}^n a_r x^r, b_r=1+\frac{a_r}{a_{r-1}}$ and $\prod\limits_{{r=1}}^n b_r=\frac{(101)^{100}}{100 !}$, then value of $n$ is

On the sides $\mathrm{AB}, \mathrm{BC}, \mathrm{CA}$ of a $\triangle \mathrm{ABC}, 3,4,5$ distinct points (excluding vertices A, B, C) are respectively chosen. The number of triangles that can be constructed using these chosen points as vertices are :

The expression ${\;^n}{C_r} + 4.{\;^n}{C_{r - 1}} + {6.^n}{C_{r - 2}} + 4.{\;^n}{C_{r - 3}} + {\;^n}{C_{r - 4}}$ equals

All possible two factors products are formed from numbers 1, 2, 3, 4,…,200. The number of factors out of the total obtained which are multiples of 5, is

A parallelogram is cut by two sets of $m$ lines parallel to its sides. The number of parallelograms thus formed is

If a polygon has 44 diagonals, then the number of its sides are

${ }^{14} {C}_7+\sum\limits_ {{i}=1}^3{ }^{17-{i}} {C}_6=$

Six points in a plane be joined in all possible ways by indefinite straight lines and if no two of them be coincident or parallel, and no three pass through the same point (with the exception of the original 6 points). The number of distinct points or intersection is equal to

The value of $\frac{1}{1 ! 50 !}+\frac{1}{3 ! 48 !}+\frac{1}{5 ! 46 !}+\ldots .+\frac{1}{49 ! 2 !}+\frac{1}{5 ! 1 !}$ is :

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then, the number of such arrangements is

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is

The number of integers which lie between 1 and ${10^6}$ and which have the sum of the digits equal to 12, is

A shopkeeper sells three varieties of perfumes and he has a large number of bottles of the same size of each variety in his stock. There are 5 places in a row in his showcase. The number of different ways of displaying the three varieties of perfumes in the show case is

Consider three boxes, each containing 10 balls labelled 1, 2, …, 10. Suppose one ball is randomly drawn from each of the boxes. Denote by ${n_i},$ the label of the ball drawn from $i$ th box, (i = 1, 2, 3). Then, the number of ways in which the balls can be chosen such that ${n_1} < {n_2} < {n_3}$ is

Find ${ }^n C_{21}$, if ${ }^n C_{10}={ }^n C_{11}$.

The value of ${\;^{50}}{C_4} + \sum \limits_{r = 1}^6 {\;^{56 - r}}{C_3}$ is

From an unlimited number of red, white, blue and green balls, a selection of 18 balls is to be made so that there are at least two of each colour. If the number of selection is $k$, then $k$ is equal to:

The results of 21 football matches (win, lose or draw) are to be predicted. The number of forecasts that contain exactly 18 correct results is

There are 10 points in a plane, out of these 6 are collinear. The number of triangles formed by joining these points is

The total number of different combinations of letters which can be made from the letters of the word MISSISSIPPI is

A person goes for an examination in which there are four papers with a maximum of m marks from each paper. The number of ways in which one can get $2m$ marks, is

If the number of ways of selecting $n$ cards out of unlimited number of cards bearing the number $0,9,3$, so that they cannot be used to write the number 903 is 93 , then $n$ is equal to

A lady gives a dinner party for six guests. The number of ways in which they may be selected from among ten friends, if two of the friends will not attent the party together, is

In a cricket championship there are 36 matches. The number of terms, if each plays 1 match with other are

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

The number of ordered pairs of non-negative integers $\{x, y\}$ having sum 7596 without any carries is equal to