Number of ways in which 7 people can occupy 6 seats, 3 seats on each side in a first class railway compartment if two specified persons are to be always include d and occupy adjacent seats on the same side is `(5!)dotk ,t h e nk` has the value equal to ___________.

The number of ways in which 7 people can sit around a table if two persons take consecutive seats is

A tea party is arranged for 16 people among two sides of a long table with 8 chairs on each side. Four men wish to sit on one particular side and two on the other side. The number of ways they can be seated is

Let theta=(a_(1),a_(2),a_(3),...,a_(n)) be a given arrangement of n distinct objects a_(1),a_(2),a_(3),…,a_(n) . A derangement of theta is an arrangment of these n objects in which none of the objects occupies its original position. Let D_(n) be the number of derangements of the permutations theta . There are 5 different colour balls and 5 boxes of colours same as those of the balls. The number of ways in which one can place the balls into the boxes, one each in a box, so that no ball goes to a box of its own colour is

In how many ways, can three girls can three girls and nine boys be seated in two vans, each having numbered seats, 3 in the and 4 at the back? How many seating arrangements are possible if 3 girls should sit together in a back row on adjacent seats? Now, if all the seating arrangements are equally likely, what is the probability of 3 girls sitting together in a back row on adjacent seats?

(i) A person went to a restaurant for dinner. In the menu card, the person saw 10 Indian and 7 Chinese food items. In how many was the person can select either an Indian or a Chinese food? (ii) There are 3 typesof toy car and 2 types of toy train available in a shop. Find the number of ways a baby can buy a toy car and a toy train? (iii) How many of two-digit numbers can be formed using 1,2,3,4,5 without repetiton of digits? (iv) There persons enter in to a conference hall in which there are 10 seats, In how many ways they can take their place? (v) In how many ways 5 persons can be seated in a row?

A class has tree teachers, Mr. X , Ms. Y and Mrs. Z and six students A , B , C , D , E , F . Number of ways in which they can be seated in a line of 9 chairs,if between any two teachers there are exactly two students is

In a three-storey building, there are four rooms on the ground floor, two on the first and two on the second floor. If the rooms are to be allotted to six persons, one person occupying one room only, the number of ways in which this can be done so that no floor remains empty is a. "^8P_6-2(6!) b. "^8P_6 c. "^8P_5(6!) d. none of these

Column I, Column II Four dice (six faced) are rolled. The number of possible out comes in which at least one dice shows 2 is, p. 210 Let A be the set of 4-digit number a_1a_2a_3a_4w h e r ea_1> a_2> a_3> a_4dot then n(A) is equal to, q. 750 The total number of three-digit numbers, the sum of whose digits is even, is equal to, r. 671 The number of four-digit numbers that can be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7 so that each number contains digit 1 is, s. 450

For examination, a candidate has to select 7 subjects from 3 different groups A, B, C which contain 4, 5, 6 subjects, respectively. The number of different way in which a candidate can make his selection if he has to select at least 2 subjects form each group is a.25 b. 260 c. 2700 d. 2800

Match each of the set on the left in the roster form with the same set on the right described in set-builder form: (i) {1, 2, 3, 6} (a) {x : x is a prime number and a divisor of 6} (ii) {2, 3} (b) {x : x is an odd natural number less than 10} (iii) {M,A,T,H,E,I,C,S} (c) {x : x is natural number and divisor of 6} (iv) {1, 3, 5, 7, 9} (d) {x : x is a letter of the word MATHEMATICS}.