Question 1

A Group consists of 4 couples in which each of the 4 persons have one wife each. In how many ways could they be arranged in a straight line such that the men and women occupy alternate positions?

Accepted Answer

Case I : MW MW MW MW Case II: WM WM WM WM Let us arrange 4 men in 4! ways, then we arrange 4 women in 4P4 ways at 4 places either left of the men or right of the men. Hence required number of arrangements 4 ! × 4 P 4 + 4 ! × 4 P 4 = 2 × 576 = 1152

Question 2

The Number of times the digit 8 will be written when listing the integers from 1 to 1000 is :

Accepted Answer

Since 8 does not occur in 1000, we have to count the number of times 8 occurs when we list the integers from 1 to 999. Any number between 1 and 999 is of the form xyz, where 0 ≤ x , y , z ≤ 9 . Let us first count the numbers in which 8 occurs exactly once. Since 8 can occur atone place in 3 C 1 ways. There are 3 x 9 x 9 such numbers. Next, 8 can occur in exactly two places in 3 C 2 × 9 such numbers. Lastly, 8 can occur in all three digits in one number only. Hence, the number of times 8 occur is 1 × 3 × 9 2 + 2 × 3 × 9 + 3 × 1 = 300

Question 3

There are 41 students in a class, number of girls is one more than number of guys. we need to form a team of four students. all four in the team cannot be from same gender. number of girls and guys in the team should NOT be equal. How many ways can …

Accepted Answer

Given G + B= 41 and B = G-1 Hence, G = 21 and B = 20 Now we have 2 options, 1G and 3M (or) 3G and 1M (2G and 2M or 0G and 4M or 4G and oM are not allowed), Total : C 1 21 × C 3 20 + C 3 21 × C 1 20 = 50540 ways.

Question 4

How many four digit even numbers can be formed using the digits {2, 7, 5, 3, 9, 1} ?

Accepted Answer

The given digits are 1, 2, 3, 5, 7, 9 A number is even when its units digit is even. Of the given digits, two is the only even digit.Units place is filled with only '2' and the remaining three places can be filled in ⁵P₃ ways. Number of even numbers = ⁵P₃ = 60.

Question 5

If it is possible to make a meaningful word with the first, the seventh, the ninth and the tenth letters of the word RECREATIONAL, using each letter only once, which of the following will be the third letter of the word? If more than one such word c…

Accepted Answer

The first, the seventh, the ninth and the tenth letters of the word RECREATIONAL are R, T, O and N respectively. Meaningful word from these letters is only TORN. The third letter of the word is ‘R’.

Question 6

In the below word how many words are there in which R and W are at the end positions? RAINBOW

Accepted Answer

When R and W are the first and last letters of all the words then we can arrange them in 5!ways. Similarly When W and R are the first and last letters of the words then the remaining letters can be arrange in 5! ways. Thus the total number of permutations = 2 x 5! = 2 x 120 = 240

Question 7

A college has 10 basketball players. A 5-member team and a captain will be selected out of these 10 players. How many different selections can be made?

Accepted Answer

A team of 6 members has to be selected from the 10 players. This can be done in 10C6 or 210 ways. Now, the captain can be selected from these 6 players in 6 ways. Therefore, total ways the selection can be made is 210×6= 1260

Question 8

In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together

Accepted Answer

In the word 'CORPORATION', we treat the vowels OOAIO as one letter. Thus, we have CRPRTN (OOAIO). This has 7 (6 + 1) letters of which R occurs 2 times and the rest are different. Number of ways arranging these letters =7!/2!= 2520. Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged in 3!/5!= 20 ways. Required number of ways = (2520 x 20) = 50400.

Question 9

In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?

Accepted Answer

Required number of ways = 7 C 5 × 3 C 2 = 63

Question 10

How many 3-digit numbers can be formed with the digits 1,4,7,8 and 9 if the digits are not repeated ?

Accepted Answer

Three digit number will have unit’s, ten’s and hundred’s place. Out of 5 given digits any one can take the unit’s place. This can be done in 5 ways. ... (i) After filling the unit’s place, any of the four remaining digits can take the ten’s place. This can be done in 4 ways. ... (ii) After filling in ten’s place, hundred’s place can be filled from any of the three remaining digits. This can be done in 3 ways. ... (iii) So,by counting principle, the number of 3 digit numbers = 5x 4 x 3 = 60

Question 11

Find the number of ways to arrange 4 people in groups of 3 at a time where order matters?

Accepted Answer

P(4,3)= P 3 4 = 24

Question 12

Find the number of ways to arrange 6 items in groups of 4 at a time where order matters?

Accepted Answer

6P4 = 6! / (6-4)! = 360

Question 13

Find the number of ways to take 20 objects and arrange them in groups of 5 at a time where order does not matter.?

Accepted Answer

C 5 20 = 15504

Question 14

How many ways are there to select a subcommittee of 7 members from among a committee of 17?

Accepted Answer

Since it does not matter what order the committee members are chosen in, the combination formula is used. Committees are always a combination unless the problem states that someone like a president has higher hierarchy over another person. If the committee is ordered, then it is a permutation. C(17,7)= 19,448

Question 15

Determine the total number of five-card hands that can be drawn from a deck of 52 cards.

Accepted Answer

When a hand of cards is dealt, the order of the cards does not matter. If you are dealt two kings, it does not matter if the two kings came with the first two cards or the last two cards. Thus cards are combinations. There are 52 cards in a deck and we want to know how many different ways we can put them in groups of five at a time when order does not matter. The combination formula is used. C(52,5) = 2,598,960

Question 16

A school has scheduled three volleyball games, two soccer games, and four basketball games. You have a ticket allowing you to attend three of the games. In how many ways can you go to two basketball games and one of the other events?

Accepted Answer

Since order does not matter it is a combination. The word AND means multiply. Given 4 basketball, 3 volleyball, 2 soccer. We want 2 basketball games and 1 other event. There are 5 choices left. C(n,r) C(How many do you have, How many do you want) C(have 4 basketball, want 2 basketball) x C(have 5 choices left, want 1) C(4,2) x C(5,1) = (6)(5) = 30 Therefore there are 30 different ways in which you can go to two basketball games and one of the other events.

Question 17

Find the number of ways to draw a straight, (suit does not matter) beginning with a 4 and ending with a 8?

Accepted Answer

There are 5 slots. __ __ __ __ __ The first slot must be a four. There are 4 ways to put a four in the first slot. There are 4 ways to put a five in the second slot, and there are 4 ways to put a six in the third slot. etc. (4)(4)(4)(4)(4) = 1024 Therefore there are 1024 different ways to produce the desired hand of cards.

Question 18

A certain marathon had 50 people running for first prize, second, and third prize.How many ways are there to correctly guess the first, second, and third place winners?

Accepted Answer

There is 1 way to correctly guess who comes in first, second, and third. There is only one set of first, second and third place winners. You must correctly guess these three people, and there is only one way to do so.

Question 19

How many number of times will the digit ‘7' be written when listing the integers from 1 to 1000?

Accepted Answer

7 does not occur in 1000. So we have to count the number of times it appears between 1 and 999. Any number between 1 and 999 can be expressed in the form of xyz where 0 < x, y, z < 9. 1. The numbers in which 7 occurs only once. e.g 7, 17, 78, 217, 743 etc This means that 7 is one of the digits and the remaining two digits will be any of the other 9 digits (i.e 0 to 9 with the exception of 7) You have 1*9*9 = 81 such numbers. However, 7 could appear as the first or the second or the third digit. Therefore, there will be 3*81 = 243 numbers (1-digit, 2-digits and 3- digits) in which 7 will appear only once. In each of these numbers, 7 is written once. Therefore, 243 times. 2. The numbers in which 7 will appear twice. e.g 772 or 377 or 747 or 77 In these numbers, one of the digits is not 7 and it can be any of the 9 digits ( 0 to 9 with the exception of 7). There will be 9 such numbers. However, this digit which is not 7 can appear in the first or second or the third place. So there are 3 * 9 = 27 such numbers. In each of these 27 numbers, the digit 7 is written twice. Therefore, 7 is written 54 times. 3. The number in which 7 appears thrice - 777 - 1 number. 7 is written thrice in it. Therefore, the total number of times the digit 7 is written between 1 and 999 is 243 + 54 + 3 = 300

Question 20

A team of 8 students goes on an excursion, in two cars, of which one can seat 5 and the other only 4. In how many ways can they travel?

Accepted Answer

There are 8 students and the maximum capacity of the cars together is 9. We may divide the 8 students as follows Case I: 5 students in the first car and 3 in the second Or Case II: 4 students in the first car and 4 in the second Hence, in Case I: 8 students are divided into groups of 5 and 3 in 8 C 3 ways. Similarly, in Case II: 8 students are divided into two groups of 4 and 4 in 8 C 4 ways. Therefore, the total number of ways in which 8 students can travel is 8 C 3 + 8 C 4 = 56 + 70 = 126.

Permutation and Combination Questions