Permutation and combination practice · Mixed difficulty

The number of bijective functions $f:\{1,3,5,7, \ldots$, $99\} \rightarrow\{2,4,6,8, \ldots \ldots, 100\}$ such that $f(3) \geq f(9) \geq f(15) \geq f(21) \geq \ldots \geq f(99)$, is _____.

Letter of the word 'WATER' is arranged in all possible ways and written as dictionary then rank of the word 'WATER' is

A boat is to be managed by eight men of whom 2 can only row on bow side and 3 can only row on stroke side, the number of ways in which the crew can be arranged, is

An examination paper contains 8 question of which 4 has 3 possible answers, 3 have 2 possible answers each and the remaining one question has 5 possible answers. The total number of possible answers to all the questions is

There are three copies, each of 4 different books. The number of ways in which they can be arranged on a shelf, is

12 boys and 2 girls are to be seated in a row such that there are atleast 3 boys between the 2 girls. The number of ways this can be done is $\lambda \times 12 !$ ! the value of $\lambda$ is

The number of squares that we can find on a chess board is

The total number of proper factors of 7875 is

The number of positive terms in the sequence $x_n=\frac{195}{4{ }^n P_n}-\frac{{ }^{n+3} P_3}{{ }^{n+1} P_{n+1}}, n \in N$ is

Five digit number divisible by 3 is formed using $0,1,2,3,4$, 6 and 7 without repetition. Total number of such numbers are

The number of ways in which 5 boys and 5 girls can sit in a row so that all girls sit together is

The no. of dissimilar terms in the expansion of $(a+b+c)^{10}$ is

The total number of selections of at least one red ball from a bag containing 4 red balls and 3 green balls, balls of the same colour being identical, is

The number of even proper divisors of 1008 is

From 3 mangoes, 4 apples and 2 oranges. The number of selections of fruits that can be made, taking at least one of each kind is

The sum of all four digit numbers that can be med using the digits $1,2,3,4$, when repetition of digits is not allowed, is

The value of $\frac{(2 n) !}{n !}$ is

If ${\;^{56}}{P_{r + 6}}:{\;^{54}}{P_{r + 3}} = 30800\;:1,$ then the value of $r$ is

The number of 3 letters words, with or without meaning which can be formed out of the letters of the word 'NUMBER'.

Statement I: When repetition of letters is not allowed is 120.

Statement II: When repetition of letters is allowed is 216.

Choose the correct option.

An $n$-digit number is a positive number with exactly $n$ digits. Nine hundred distinct $n$-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of $n$ for which this is possible, is

Two teams are to play a series of 5 matches between them. A match ends in a win or loss or draw for a team. A number of people forecast the result of each match and no two people make the same forecast for the series of matches. The smallest group of people in which one person forecasts correctly for all the matches will contain $n$ people, where $n$ is

We are required to form different words with the help of the letters of the word INTEGER. Let $m_1$ be the number of words in which I and N are never together and $m_2$ be the number of words which begin with I and end with R, then $\frac{m_1}{m_2}$ is given by

The number of five digit numbers that can be formed by using digits 1, 2, 3 only, such that three digits of the formed number are identical, is

The number of arrangements that can be formed out of GANESHPURI so that "The vowels are always together is

The number of words that can be formed from the letters $a, b, c, d, e, f$, taken 3 at a time, each word containing at least one vowel is

The number of integers, greater than 7000 that can
be formed, using the digits 3, 5, 6, 7, 8 without
repetition, is

The range of the function $f(x) = {}^{7-x} P_{x-3}$ is

If $n=2^{100} 3^2$ and $\left\{d_1, d_2, \ldots d_k\right\}$ is the set of all divisors of $n$, then $\sum\limits_{{j=1}}^k \frac{1}{d_j}$ equals

The number of divisors of $n=2^7 \cdot 3^5 \cdot 5^3$, which are of the form $4 t+1(t \in N)$, is

The total number of four-digit numbers having all different digits is equal to