Permutation and Combination | Set-2

Question 1: How many number greater than ten thousand can be formed from the digits 0, 1, 2, 3, 4 without repetition ? 
Solution : In order to form a number greater than 10000 we should have only 5 digits. Since, we have 5 digits we cannot take 0 in starting position. 
_ _ _ _ 
For first digit, we have 4 choices. 
For second digit, again we have 4 choices because we can include 0 from here onwards. 
For third digit, we have 3 choices. 
and for fourth digit only left 2 choices. 
Total numbers = 4x4x3x2= 96 
Hence, only 96 numbers possible. 

Question 2: In how many ways can 4 boys and 4 girls be seated around a circular table so that no two boys are in adjacent positions? 
Solution : If we first put 4 boys around the table, we can do this in 3! ways. 
Once the 4 boys are placed, we have to place 4 girls around the same table. 
Now, we can see 4 vacant places are there between all 4 boys so we can do in 4! ways. 
Total number of sitting arrangement = 3! x 4! 

Question 3: Out of the 11 points in a plane, 4 are collinear. How many straight line can be formed ? 
Solution : If all points were non-collinear then possible lines would have been ¹¹C₂. 
But, 4 points are collinear lie on the same line. So, they are all counted as a single line. 
Total number of straight line = ¹¹C₂ - ⁴C₂ + 1 
= 11x10/2 - 4x3/2 + 1 
= 55 - 6 + 1 
= 50 

Question 4: Twenty people attend a party and shake hands with one another.In how many ways hand shake is possible? 
Solution : All people shake hands with one another except himself. 
1st person has 19 hand shakes and 2nd also has 19 hand shakes .........and so on 
20 people x 19 hand shakes 
and we know A handshake with B or B handshake with A, it is counted as 1 handshake. 
So, total number of handshake = 20x19/2 = 190 
Shortcut : 
for n people there are always ⁿC₂ handshakes. 

Question 5: How many different sums of money can be formed from the four type of notes Rs 10, Rs 20, Rs 50 and Rs 100 ? 
Solution : Type of notes = 4 
So, total number of sum can be formed = 2⁴ - 1 = 15 

Question 6: Five chocolates of different flavours are to be distributed in three different children such that any child get at least 1 chocolate. What is the maximum number of different ways in which this can be distributed? 
Solution : Acc. to question 
Chocolates can be distributed as [(3, 1, 1)(1, 3, 1)(1, 1, 3)] 
or [(2, 2, 1)(2, 1, 2)(1, 2, 2)] 
Total number of ways = 3 x ⁵C₃ x ²C₁ x ¹C₁ + 3 x ⁵C₂ x ³C₂ x ¹C₁ 
= 60 + 90 
= 150 

Question 7: Ram and his wife Sita both have five friends each. Ram has 2 boys and 3 girls. Sita has 3 boys and 2 girls. In how many maximum number of different ways can they invite 2 boys and 2 girls such that two of them are Ram's friend and two are Sita's friend? 
Solution : Selection can be done like that 
i) 2 boys from Ram's friends and 2 girls from his wife Sita's friends OR 
ii) 1 boy and 1 girl from Ram's friends and 1 boy and 1 girl from Sita's friends OR 
iii) 2 boys from his wife Sita's friends and 2 girls from Ram's friends 
Total number of ways = ²C₂.²C₂ + ³C₁ x ²C₁ x ³C₁ x ²C₁ + ³C₂.³C₂ 
= 1 + 36 + 9 
= 46 ways 

Question 8: In how many ways can 4 notebooks can be distributed to 5 students if each can get any number of notebooks? 
Solution : Since all the notebooks are identical or distinct we don't know. 
So, we take all are distinct and it can be distributed in 5⁴ ways. 

Question 9: How many batting orders are possible for the Indian cricket team if there is a squad of 16 to choose from such that Virat Kohli and Rohit Sharma are always chosen? 
Solution : We need to select 9 players out of 14 players because two of them is already selected. 
The selection of 11 players can be done in¹⁴C₉ ways. 
But batting order is also required to calculate for these 11 players so arrangements can be done in 11! ways. 
Total number of batting orders possible = ¹⁴C₉ . 11! 

Question 10: How many motor vehicle registration numbers plates can be formed for the state Haryana having code like (HR 12Q 8702) with the digits 0, 1, 2, 3, 4, 5, 6 and contains consonant at the alphabetical place(No digits being repeated). 
Solution : For Haryana state number plate always contains HR in starting. 
Two consonants already used in HR so remaining consonant = 21 -2 = 19. 
So, possible number of plates = 1(HR)x7x6x19(consonants)x5x4x3x2 = 95760