1. An executive committee of 6 is to be formed from 4 ladies
and 7 gentlemen. In how many ways can this be formed when the committee
contains (i) only 2 lady members, (ii) at least 2 lady members?
2. Find the number of committees of 5 members that can be
formed from 6 gentlemen and 4 ladies if each committee has at least one lady
and two gentlemen.
3. A committee of 5 is to be formed from six ladies and four
gentlemen. In how many ways this can be done so that the committee contains (i)
exactly two ladies, (ii) at least two ladies, (iii) at most two ladies?
4. In a cricket team of 14 players 6 are bowlers. How many
different teams of 11 players can be selected keeping at least 4 bowlers in the
team?
5. A box contains 12 lamps of which 5 are defective. In how
many ways can a sample of 6 be selected at random from the box so as to include
at most 3 defective lamps?
6. An examinee is required to answer 6 questions out of 12
questions which are divided into two groups each containing 6 questions, and he
is not permitted to answer more than 4 questions from any group. In how many
ways can he answer 6 questions?
7. A question paper contains 10 questions, which are divided
into two groups each containing 5 questions. A candidate is asked to answer 6
questions only, and to choose at least 2 questions from each group. In how many
different ways can the candidate make up his choice?
8. In how many ways can a team of 11 cricketers be chosen
from 9 batsmen and 6 bowlers to give a majority of batsman if at least 3 bowlers
are to be included?
9. The Indian Cricket Eleven is to be selected out of
fifteen players, five of them are bowlers. In how many ways the team can be
selected so that the team contains at least three bowlers?
10. How many combinations can be formed of eight counters
marked 1, 2, 3, 4, 5, 6, 7, 8 taking them 4 at a time, there being at least one
odd and one even counter in each combination?
11. Find the number of permutations of the letters of the
words FORECAST and MILKY taking 5 at a time of which 3 letters from the first
word and 2 from the second.
12. In how many ways can the crew of an eight-oared boat be
arranged if 2 of the crew can row only on the stroke side and 1 can row only on
the bow side?
13. Of the 17 articles, 12 are alike and the remaining 5 are
different. Find the number of combinations, if 13 articles are taken at a time.
14. Out of 3n given things 2n are alike and the rest are
different. Show that a selection of 2n things can be made from these 3n things
in 2" different ways.
15. Show that there are 136 ways of selecting 4 letters from
the word EXAMINATION.
16. Find the total number of ways of selecting 5 letters
from the letters of the word INDEPENDENT.
17. (i) Find the number of combinations in the letters of
the word STATISTICS taken 4 at a time.
(ii) Find the
number of permutations in the letters of the word PROPORTION taken 4 at a time.
18. How many different numbers of 4 digits can be formed
with the digits 1, 1, 2, 2, 3, 4?
19. (i) From 4 apples, 5 oranges mangoes, how many
selections of fruits can be made, taking at least one of each kind if the
fruits of the same kind are of different shapes?
(ii) In how many ways can one or more fruits be selected
from 4 apples, 5 oranges and 3 mangoes, if the fruits of the same kind be of
the same shape?
20. Find the total number of combinations taking at least
one green ball and one blue ball, from 5 different green balls, 4 different
blue balls and 3 different red balls.
21. How many different algebraic quantities can be formed by
combining a, b, c, d, e with + and - signs, all the letters taken together?
22. There are n points in space, no four of which are in the
in the same place with the exception of m points, all of which are in the same
plane. How many planes can be formed by joining them?
23. n1, n2 and n3,
points are given on the sides BC, CA and AB respectively of the triangle ABC.
Find the number of triangles formed by taking these given points as vertices of
a triangle.
24. A man has 7 relatives, 4 of them are ladies and 3
gentlemen; his wife has also 7 relatives, 3 of them are ladies and 4 are
gentlemen. In how many ways can they invite dinner party of 3 ladies and 3
gentlemen so that there are 3 of the man's relatives and 3 of the wife's
relatives?
25. Eighteen guests have to be seated, half on each side of long table. Four particular guests desire to sit on one particular side and three others on the other side Determine the number of ways in which the arrangements can be made.
