Permutations and Combinations · Question 21

From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include at least 4 batsmen and at least 4 bowlers. One batsman and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is

1. 145
2. 165
3. 155
4. 135

For n 2, let Sn denote the set of all subsets of \1, 2, , n\ with no two consecutive numbers. For example \1, 3, 5\ S6, but \1, 2, 4\ S6. Then n(S5) is equal to

Let m and n, (m<n), be two 2-digit numbers. Then the total number of pairs (m, n), such that (m, n)=6, is .

All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number n be denoted by Wn. Let the probability P(Wn) of choosing the word Wn satisfy P(Wn) = 2 P(Wn-1), n > 1. If P(CDBEA) = 2 2 -1, , N, then + is equal to

If the number of seven-digit numbers, such that the sum of their digits is even, is m n 10n; m, n \1, 2, 3, , 9\, then m + n is equal to

Line L1 of slope 2 and line L2 of slope 12 intersect at the origin O. In the first quadrant, P1, P2, , P12 are 12 points on line L1 and Q1, Q2, , Q9 are 9 points on line L2. Then the total number of triangles, that can be formed having vertices at three of the 22 points O, P1, P2, , P12, Q1, Q2, , Q9, is:

1. 1026
2. 1188
3. 1134
4. 1080

The largest n N such that 3n divides 50 ! is :

1. 22
2. 20
3. 21
4. 23

The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to :

1. 360
2. 2520
3. 1820
4. 45

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is : [figure]

1. 5880
2. 840
3. 960
4. 5760

Let P be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in P are formed by using the digits 1, 2 and 3 only, then the number of elements in the set P is :

1. 164
2. 158
3. 161
4. 173