PERMUTATIONS AND COMBINATIONS - 3 (Simple Applications on and )
Important Results
1. Total number of selections of one or more objects from
2. Total number of selections of any number of things from
3. Total number of selections from
4. Total number of selections of
5. Total number of ways to divide
Results on distribution
6. Distribution of
Given | Condition | Number of ways |
---|---|---|
Empty boxes are allowed | ||
Empty boxes are not allowed | coefficient of |
|
Empty boxes are allowed | ||
Empty boxes are not allowed |
Division of items into groups
7. (i) Groups of unequal size.
- Number of ways in which
items can be divided into unequal groups containing items is - Number of ways to distribute
items among 3 persons in the group containing items is !
(ii) Groups of equal size
- Number of ways in which
different items can be divided equally into groups each containing n objects
Note
(i) If there are
(ii) If there are
8. Results related with points, Lines, Rectangle, Polygon, Circle, etc.
(i) If these are
(ii) Number of points
(iii) Number of diagonals in a regular polygon having
(iv) Number of parallelograms when a parallelogram is cut by two sets of
(v)

(vi)

(vii) Maximum number of parts in which a plane can be divided by
(viii) Maximum number of points of intersection of
(ix) Maximum number of points of intersection of
(x) Maximum number of points of intersection of
De-arrangement
Number of arrangement of
Exponent of prime in !
Solved examples
1. The number of zeroes at the end of 100 ! is
(a) 23
(b) 24
(c) 25
(d) None of these
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Solution:
Answer: (b)
2. The total number of integral solutions of the triplet
(a) 30
(b) 60
(c) 120
(d) None of these
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Solution:
30 positive integral solutions
Total number of integral solutions with negative integers included is
Answer: (c)
3. The total number of squares in a chess board is
(a) 64
(b) 65
(c) 204
(d) None of these
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Solution:
Answer: (c)
4.
(a) 210
(b) 211
(c) 212
(d) None of these
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Solution:
Answer: (b)
5. The number of quadrilaterals that can be formed using 10 points in a plane out of which 4 are collinear is
(a) 210
(b) 209
(c) 185
(d) None of these
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Solution:
Answer: (c)
6. The total number of distinct rational numbers
(a) 15
(b) 13
(c) 11
(d) None of these
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Solution:
Values of |
Possible rational numbers |
---|---|
1 | |
2 | |
3 | |
4 | |
5 |
Out of 15 possible rational numbers, only 11 are distinct.
Answer: (c)
7. The sum of 5 digit number in which only odd digits occur without repetition is
(a) 277775
(b) 555550
(c) 1111100
(d) None of these
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Solution:
Sum of n digit numbers
Answer: (d)
Practice questions
1. An n digit number is a positive number with exactly
(a) 6
(b) 7
(c) 8
(d) 9
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Answer: (b)2. Match the following:
Consider all possible permutations of the letters of the word ENDEANOEL
Column I | Column II |
---|---|
(a) The number of permutations containing the word ENDEA is | (p) |
(b) The number of permutations in which the letteE occurs in the first and last position is | (q) |
(c) The number of permutations in which none of the lette |
(r) |
(d) The number of permutations in which the letter |
(S) |
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Answer: a3. Five balls of different colors are to be placed in 3 boxes of different sizes. Each box can hold all 5 balls. The number of ways we can place the balls so that no box is empty, is
(a) 116
(b) 126
(c) 144
(d) 150
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Answer: (d)4. A student is allowed to select atmost
(a) 3
(b) 4
(c) 6
(d) 5
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Answer: (a)5. A rectangle with sides
(a)
(b)
(c)
(d)
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Answer: (c)6. Out of 5 apples, 10 mangoes and 15 oranges, the number of ways of distributing 15 fruits each to two persons, is
(a) 56
(b) 64
(c) 66
(d) 72
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Answer: (c)7. Match the following
Column I | Column II |
---|---|
(a). The number of positive integral solutions of the equati |
(p). 3 |
(b). Let be the element of the set |
(q). 4 |
(c). Let a be a factor of 120 . If |
(r). 5 |
(s). 8 | |
(t). 16 |
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Answer: a8. The maximum number of points into which 4 circles & 4 straight lines intersect is
(a) 26
(b) 50
(c) 56
(d) 72
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Answer: (b)9. A is a set containing
(a)
(b)
(c)
(d) None of these
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Answer: (d)10. Number of points having position vector
(a) 70
(b) 140
(c) 210
(d) 280
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Answer: (a)11. Read the passage and answer the following questions.
(i)
(a)
(b)
(c)
(d) None of these
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Answer: (a)(ii)
(a)
(b)
(c)
(d) None of these
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Answer: (c)(iii)
(a)
(b)
(c)
(d) None of these