Combinations practice · Medium

For $x \in {R}$, let $[x]$ denote the greatest integer $\leq x$, then value of $\left[-\frac{1}{3}\right]+\left[-\frac{1}{3}-\frac{1}{100}\right]$ $+\left[-\frac{1}{3}-\frac{2}{100}\right]+\ldots+\left[-\frac{1}{3}-\frac{99}{100}\right]$ is

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then, the number of ways it can be done is

${ }^{n-1} C_r=\left(k^2-8\right)^n C_{r+1}$ if and only if:

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

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

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:

If $\sum\limits_{r = 0}^{25} {\left\{ {{}^{50}{C_r} \cdot {}^{50 - r}{C_{25 - r}}} \right\} = K\left( {{}^{50}{C_{25}}} \right)}$. Then, $K$ is equal to

In a shop there are five types of ice-creams (each type is available unlimited). 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 $6 {A}^{\prime} \mathrm{s}$ and $4 {B}$ 's in a row.

A dice is rolled 4 times, the numbers appearing on the dice in different throws are listed, such that the largest number appearing in the list is not 4 , in how many ways this can be done

The sum $\sum_{r=1}^{10}\left(r^{2}+1\right) \times(r !)$ is equal to :

If the number of 5-element subsets of the set $A=\left\{a_{1}, a_{2}, \ldots, a_{20}\right\}$ of 20 distinct elements is $k$ times the number of 5 -element subsets containing $a_{4}$, then $k$ is

If ${ }^n C_r$ denotes the number of combinations of $n$ things taken $r$ things at a time, then the expression ${ }^n C_{r+1}+{ }^n C_{r-1}+2{ }^n C_r$ is

The number of all the odd divisors of 3600 is

In an examination a candidate has to pass in each of the papers to be successful. It the total number of ways to fail is 63 , how many papers are there in the examination?

There are $n$ straight lines in a plane, no two of which are parallel and no three pass through the same point. Their points of intersection are joined. Then, the number of fresh lines thus obtained is

Determine $n$ if ${ }^{2 n} {C}_3:{ }^n {C}_2=12: 1$

Consider the following statements.

I. ${ }^n C_r+2 \, { }^n C_{r-1}+{ }^n C_{r-2}={ }^{n+2} C_r$

II. ${ }^n C_p={ }^n C_q \Rightarrow p=q$ or $p+q=n$

Choose the correct option.

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is

If $\left({ }^{30} C_{1}\right)^{2}+2\left({ }^{30} C_{2}\right)^{2}+3\left({ }^{30} C_{3}\right)^{2}+\ldots+30\left({ }^{30} C_{30}\right)^{2}$ $=\frac{\alpha 60 !}{(30 !)^{2}}$ then $\alpha$ is equal to :

${ }^m C_{r+1}+\sum\limits_{{k=m}}^n{ }^k C_r$ is equal to

Statement-1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is ${ }^{9} C_{3}$.
Statement-2: The number of ways of choosing any 3 places from 9 different places is ${ }^{9} C_{3}$.

If $m$ denotes the number of 5 digit number if each successive digits are in their descending order magnitude and ${n}$ is the corresponding figure when the digit are in their ascending order of magnitude

The number of different ways in which five alike dashes and eight alike dots can be arranged using only seven of these 'dashes' and 'dots' is equal to