Q.
A football stadium has four entrance gates and nine exits. In how many different ways can a man enter and leave the stadium?
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No of entrance = 4
No of exits = 9
∴ Total no of ways of entering and leaving the stadium = 4 × 9 = 36
Q.
There are six doors in a hostel. In how many ways can a student enter the hostel and leave by a different door?
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No. of doors in a hostel = 6
So, no. of doors to enter the hostel = 6
Since, student has to exit by different door, no. of doors to exit the hostel = 6 - 1 = 5
∴ Total ways to enter and leave by different door = 6 × 5 = 30
Q.
In how many ways can a man send three of his children to seven different colleges of a certain town?
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Total no. of colleges = 7
Total no. of children = 3
For 1st child no. of college = 7
For 2nd child no. of college = 7 - 1 =6
For 3rd child no. of college = 6 - 1 = 5
∴ Total no. of ways of sending his children to different colleges = 7 × 6 × 5 = 210
Q.
Suppose there are five main roads between the cities A and B. In how many ways can a man go from a city to the other and return by a different road?
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Total no. of roads between the cities A and B = 5
No. of roads to go from city A to B = 5
No. of different roads left to return from city B to A = 4
∴ Total no. of ways to go from A to B and return by different road = 5 × 4 = 20
Q.
There are five main roads between the cities A and B and 4 between B and C. In how many ways can a person drive from A to C and return without driving on the same road twice?
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No. of roads between A and B = 5
No. of roads between B and C = 4
Since 1 road each from A to B and B to C will be used for going, only 3 and 4 roads will be left while returning from C to B and B to A respectively.
∴ Total no. of ways of travelling from A to C and returning by different road = 5 × 4 × 3 × 4 = 240
Q.
How many numbers of at least three different digits can be formed from the integers 1, 2, 3, 4, 5, 6?
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Total no of digits = 6
No. of 3 digits numbers formed = 6 × 5 × 4 = 120
No. of 4 digits numbers formed = 6 × 5 × 4 × 3 = 360
No. of 5 digits numbers formed = 6 × 5 × 4 × 3 × 2 = 720
No. of 6 digits numbers formed = 6 × 5 × 4 × 3 × 2 × 1 = 720
∴ Total no. of numbers formed = 120 + 360 + 720 + 720 = 1,920
Q.
How many numbers of three digits less than 500 can be formed from the integers 1, 2, 3, 4, 5, 6?
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We have 3 digits numbers to be formed.
For the number to be less than 500, the hundred place can have only 1 or 2 or 3 or 4 integers.
No. of integers at hundred place = 4
No. of integers at tenth place = 4
No. of integers at one place = 3
∴ Total no. of numbers formed = 4 × 4 × 3 = 48
Q.
Of the numbers formed by using all the figures 1, 2, 3, 4, 5 only once, how many are even?
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For the numbers to be even, the one place can have the integer either 2 or 4.
No. of integers at one place = 2
No. of integers at tenth place = 4
No. of integers at hundred place = 3
No. of integers at thousands place = 2
No. of integers at ten-thousanth place = 1
∴ Total no. of numbers formed = 2 × 4 × 3 × 2 × 1 = 48
Q.
How many numbers between 4000 and 5000 can be formed with the digits 2, 3, 4, 5, 6, 7?
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To form the number between 4000 and 5000, thousanth place can have only integer 4 i.e., only one number.
No. of intergers at thousandth place = 1
No. of intergers at hundred place = 5
No. of integers at tenth place = 4
No. of integers at one place = 3
∴ Total no. of numbers formed = 1 × 5 × 4 × 3 = 60
Q.
How many numbers of three digits can be formed from the integers 2, 3, 4, 5, 6? How many of them will be divisible by 5?
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No. of 3 digit numbers formed = 5 × 4 × 3 = 60
No. of numbers divisible by 5 = 1 × 4 × 3 = 12
