Permutation and combination practice · Mixed difficulty

If the permutations of $a, b, c, d$ and $e$ taken all together are written down in alphabetical order as in dictionary and numbered, the rank of the permutation debac is

Find the number of ways in which 8 different flowers can be strung to form a garland so that 4 particular flowers are never separated

If ${ }^{(m+n)} P_2=90$ and $^{(m-n)} P_2=30$, then $(m, n)$ is

If $0 < r < s \leq n$ and ${ }^n P_r={ }^n P_s$, then value of $r+s$ is

$\left({ }^8 {C}_1-{ }^8 {C}_2+{ }^8 {C}_3-{ }^8 {C}_4+{ }^8 {C}_5-{ }^8 {C}_6+{ }^8 {C}_7-{ }^8 {C}_8\right)$ equals:

If ${ }^n C_3+{ }^n C_4 > { }^{n+1} C_3$, then

The number of proper divisors of 2520 is

The number of ways of putting 5 identical balls in 10 identical boxes, if not more than one can go in a box is

There are 10 true-false questions in an examination. Then, these questions can be answered in

The products of any $r$ consecutive natural numbers is always divisible by

The value of $\left( {{{2.}^1}{P_0} - {{3.}^2}{P_1} + {4^3}{P_2}......\mathrm{upto}{\mkern 1mu} {\mkern 1mu} \mathrm{51th}{\mkern 1mu} {\mkern 1mu} \mathrm{term}} \right)\mathrm{ + }\left( {\mathrm{1}! - \mathrm{2}!\mathrm{ + 3}!...\mathrm{upto}{\mkern 1mu} {\mkern 1mu} \mathrm{51th}{\mkern 1mu} {\mkern 1mu} \mathrm{term}} \right)$ is equal to

The number of integral points that lie exactly in the interior of the triangle with vertices $(0,0),(0,21),(21,0)$, is

If number of ways of arranging letters aaaabbbbccdd in a line so that there should be at least one b in between two a&rsquo;s is $k,$ then the value of $\left| {15000 - k} \right|$ is

Four boys picked up 30 mangoes. The number of ways in which they can divide them if all mangoes be identical, is

Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls, is

If $n$ objects are arranged in a row, then the number of ways of selecting three objects, so that no two of them are next to each other, is

If ${ }^{{n}+2} C_3={ }^{{n}+3} P_2-20$, then $n$ is equal to:

The number of ways in which 3 boys and 3 girls (all are of different heights) can be arranged in a line, so that boys as well as girls among themselves are in decreasing order of height from left to right, is

Number of positive integer $n$ less than 17, for which $n !+(n+1) !+(n+2)!$ is an integral multiple of 49, is

If $2{\;^{n + 1}}{P_{n - 1}}:{\;^{2\;n - 1}}{P_n} = 3\;:5,$ then $n$ is equal to

If $l=\operatorname{LCM}$ of 8 !, 10 ! and 12 ! and $h=\operatorname{HCF}$ of 8 !, 10 ! and 12 !, then $\frac{l}{h}$ is equal to

Let $n=1 !+4 !+7 !+\cdots+400$ !. Then ten's digit of $n$ is

The total number of factors (excluding 1) of 2160 is

Number of 6 digit numbers which can be formed if the sum of their digits has to be 51 , is

The sum of all that can be formed with the digits 2,3,4,5 taken all at a time is

The number of distinct rational numbers $x$ such that $0 < x < 1$ and $x=\frac{p}{q}$, where $p, q \in\{1,2,3,4,5,6\}$ is

If $a$ denotes the number of permutations of $x + 2$ things taken all at a time, $b$ the number of permutations of $x$ things taken 11 at a time and $c$ the number of permutations of $x - 11$ things taken all at a time such that $a = 182\;bc$, then the value of $x$ is

Total number of positive integral solutions of $x_1 \cdot x_2 \cdot x_3=60$ is equal to

A class has $n$ students. We have to form a team of the students including atleast two students and also excluding atleast two students. The number of ways of forming the team is

The total number of flags with three horizontal strips, in order that can be formed using 2 identical red, 2 identical green and 2 identical white strips, is equal to