Permutation and combination practice · Mixed difficulty

Consider the following statement:

1.The number of ways of arranging m different things taken all at a time in which $p \leqslant m$ perticular things are never together is $m! - \left( {m - p + 1} \right)!p!$

2. A pack of 52 cards can be divided equally among four players in order in $\frac{{52!}}{{{{\left( {13!} \right)}^4}}}$ ways

Which of these is/are correct?

The number of ways distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

We are to form different words with 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 ${m_1}/{m_2}$ is equal to

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

Ten persons numbered $1,2,3, \ldots, 10$ play a chess tournament, each player playing against every player exactly one game. It is known that no game ends in a draw. Let $w_1, w_2, w_3, \ldots, w_{10}$ be the number of games won by player $1,2,3, \ldots, 10$ respectively and $l_1, l_2, l_3, \ldots$, $l_{10}$ be the number of games lost by the players $1,2,3, \ldots$, 10 respectively, then

The sum of the digits in the unit place of all the numbers formed with the help of $3,4,5,6$ taken all at a time is

Number of numbers greater than 24000 that can be formed by using $1,2,3,4,5$ when no digit being repeated is

Let $\mathrm{A}$ be the set of all positive prime integers less than 30 . The number of different rational numbers whose numerator and denominator belongs to $\mathrm{A}$ is

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48 , isThe number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48 , is

$n$ different toys have to be distributed among $n$ children. Total number of ways in which these toys can be distributed, so that exactly one child gets no toy, is equal to

If letters of the word 'KUBER' are written in all possible orders and arranged as in a dictionary, then rank of the word 'KUBER' will be

The one's and ten's digits in $1 !+2 !+3 !+\ldots+49$! are respectively

A man has three friends. The number of ways he can invite one friend everyday for dinner on six successive nights, so that no friend is invited more than three times, is

The no. of ways in which 5 boys and 4 girls sit around a circular table so that no two girls sit together is

The number of arrangements which can be made out of the letters of the word ALGEBRA, without changing the relative order (positions) of vowels and consonants, is

8 players $\mathrm{S}_1, \mathrm{~S}_2, \ldots . . . \mathrm{S}_8$, play in tournament. Number of ways in which they can be grouped into 4 pairs so that $S_1$ and $S_2$ are in different group is

Eleven books consisting of 5 Mathematics, 4 physics and 2 Chemistry are places on a shelf. The number of possible ways of arranging them on the assumption that the books of the same subject are all together, is

If the letters of the word ‘SACHIN’ are arranged in all possible ways and these words are written out as in dictionary, then the word ‘SACHIN’ appears at serial number

The number of ways in which $n$ things of which $r$ are alike, can be arranged in a circular order, is

Four married couples are to be seated in a row having 8 chairs. The number of ways so that spouses are seated next to each other, is

The value of $\sum \limits_{r = 1}^n \,\frac{{{\;^n}{P_r}}}{{r!}}$ is

A man has 6 friends. In how many ways he can invite one or more of them at dinner?

Total no. of ways in which six ' + ' and four '-' signs can be arranged in a row so that no two '-' signs to be together, is

Find the number of ways of arranging the letters of the word ENGINEERING

From 7 gentlemen and 4 ladies, a committee of 5 is to be formed. The number of ways in which this can be done so as to include atleast one lady is

If one test (on screening paper basis) was conducted on Batch A, maximum number of marks is $(90 \times 3)=270.4$ students get the marks lower than 80 . Coaching institute decide to inform their guardians, that is why their result card were sent to their home. The number of ways, in which all the letters were put in wrong envelopes, is

How many 10 digit numbers can be written by using the digits 1 and 2?

The number of ways 6 different flowers can be given to 10 girls, if each can receive any number of flowers is

If 20 lines are drawn in a plane such that no two of them are parallel and no three are concurrent. The number of points in which they intersect each other is

The number of ways 12 different books can be distributed equally among 4 persons, is