permuation and combination ex. 1.1

 

Chapter 1

Permutation and Combination

ex.

1.1                      

1.

Soln:

A man can enter the stadium in 4 ways.

 Again the man can leave the stadium in 9 ways.

So, total no.of ways with which a man enters and then leaves the stadium 

= 4 * 9 = 36ways.

 

2.

Soln:

There are 6 choices for a student to enter the hostel and 5 choices  to leave the hostel 

as different door is to be used.

So, total no.of ways 

= 6 * 5 = 30.

 

3.

Soln:

There are 7 choices for 1st son, 6 choices for 2nd son and 5 choices for 3rd son.

Now, by the basic principle of counting, 

the total number of ways of choice = 7 * 6 * 5 = 210.

 

4.

Soln:

A man can go from city A to city B in 5 ways and return by a different road

 city B to city A in 4 ways.

So, total no.of ways by which a man can go from city A to city B and

 returns by a different road 

= 5 * 4 = 20 ways.

 

5.

Soln:

A person can go from city A to city B in 5 ways. Again, he can go from city B to city C in 4 ways. So, a person can go from city A to city C in 5 * 4 = 20ways. The person has to return from C to A without driving on the same road twice, So, he can return from city C to city B in 3 ways and from city B to city A in 4 ways.

So, he can return from city C to city A in 3 * 4 = 12 ways.

So, Total no.of ways by which a person can go from city A to city C and return from city C to city A = 20 * 12 = 240 ways.

 

6.

Soln:

Numbers formed should be of at least 3 digits means they may be of 3 digits, 4 digits, 5 digits or 6 digits.

There are 6 choices for digit in the units place. There are 5 and 4 choices for digits in ten and hundred’s place respectively.

So, total number of ways by which 3 digits numbers can be formed = 6.5.4 = 120

Similarly, the total no.of ways by which 4 digits numbers can be formed = 6.5.4.3 = 360.

the total no. of ways by which 5 digits numbers can be formed = 6.5.4.3.2 = 720.

The total no.of ways by which 4 digits numbers can be formed = 6.5.4.3.2.1 = 720.

So, total no.of ways by which the numbers of at least 3 digits can be formed = 120 + 360 + 720 + 720 = 1920.

 

7.

Soln:

The numbers formed must be of three digits and less than 500, so the digit in the hundred’s place should be 1,2,3 or 4. So, there are 4 choices for the digit in the hundred’s place. There are 5 choice for the digit in the ten’s place. There are 4 choices for the digit in the unit’s place.

So, no of ways by which 3 digits numbers les than 500 can be formed = 4.5.4 = 80.

 

8.

 

Soln:

The numbers formed should be even. So, the digit in the unit’s place must be 2 or 4. So, the digit in unit’s place must be 2 or 4. So, for the digit in unit’s place, there are 2 choices. So, after fixing the digit in the unit’s place, remaining 4 figures can be arranged in P(4,4) ways.

Ie. (4)!(4−4)!$\frac{\left(4\right)!}{\left(4-4\right)!}$ = 4!0!$\frac{4!}{0!}$ = 4∗3∗2∗11$\frac{4\ast 3\ast 2\ast 1}{1}$ = 24 ways.

So, total no.of ways by which 5 even numbers can be formed = 2 * 24 = 48.

 

9.

Soln:

The numbers formed must be of 4 digits. The digit in the thousand’s place must always be 4. For this, there is only one choice. After that, n = 6 – 1 = 5, r = 4 – 1 = 3. Then remaining 5 figures can be placed in remaining 3 places in:

 

Or, P(5,3) ways = 5!(5−3)!$\frac{5!}{\left(5-3\right)!}$ = 5!2!$\frac{5!}{2!}$ = 5∗4∗3∗2∗12∗1$\frac{5\ast 4\ast 3\ast 2\ast 1}{2\ast 1}$ = 60 ways.

So, Total no.of ways by which 4 digits numbers between 4,000 and 5,000 can be formed = 1 * 60 = 60.

 

10.

Soln:

For the three digits numbers, there are 5 ways to fill in the 1st place, there are 4 ways to fill in the 2nd place and there are 3 ways to fill in the 3rd place. By the basic principle of counting, number of three digits numbers = 5 * 4 * 3 = 60.

Again, for three digit numbers which are divisible by 5, the number in the unit place must be 5. So, the unit place can be filled up in 1 way. After filling up the unit place 4 numbers are left. Ten’s place can be filled up in 4 ways and hundredths place can be filled up in 3 ways. Then by the basic principle of counting, no.of 3 digits numbers which are divisible by 5 = 1 * 4 * 3 = 12.