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Chapter 5: Problem 433

10 person are to be arranged around a round table. 3 persons wish to sit as a group number of ways the arrangement can be made is (a) \(9 ! \times 3 !\) (b) \(8 ! \times 3 !\) (c) \(7 ! \times{ }^{8} \mathrm{P}_{3}\) (d) \(7 ! \times 3 !\)

Short Answer

Expert verified
The arrangement of 10 people around a round table with 3 specific persons wishing to sit together as a group can be made in \(7 ! \times 3 !\) ways.

Step by step solution

01

Arrange 7 normal persons and 1 group of people around the table

Let's consider the group of 3 persons as a single entity. This makes it a total of 8 entities (7 normal persons + 1 group) to be arranged around the table. Since the round table arrangement will be considered the same if it is rotated together, we will have a total of (8-1)! = 7! ways to arrange these entities.
02

Arrange 3 persons within their group

Now, within the group of 3 persons, they can be arranged among themselves in 3! ways since there are 3 seats available within the group.
03

Calculate the total number of ways for the arrangement

To calculate the total number of ways for the arrangement, we need to multiply the number of ways from Step 1 and Step 2. So the total number of ways = (7!) × (3!). The arrangement can be made in 7! × 3! ways. The correct answer is: (d) \(7 ! \times 3 !\)

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Most popular questions from this chapter

The number of numbers greater than 3000 , which can be formed by using the digits \(0,1,2,3,4,5\) without repetition is (a) 1240 (b) 1280 (c) 1320 (d) 1380

A round table conference is to be held among 20 delegates of 20 countries. The no. of ways they can be seated if two particular delegates are never sit together is. (a) \(17 \cdot 18 !\) (b) \(18 \cdot 19 !\) (c) \((20 ! / 2)\) (d) \(19 ! \cdot 2\)

In how many ways can 15 members of a school sit along a circular table, when the secretary is to sit on side of the principal and the deputy secretary on the other side ? (a) \(2 \times 12 !\) (b) 24 (c) \(2 \times 15 !\) (d) \(2 ! \times 13 !\)

A man has 7 relative, 4 of them ladies and 3 gentlemen. His wife also have 7 relatives. 3 of them ladies and 4 gentlemen, They invite for a dinner partly 3 ladies and 3 gentlemen so that there are 3 of the men's relative and 3 of the wife's relative. The number of ways of invitation is (a) 854 (b) 585 (c) 485 (d) 548

The total number of permutations of \(\mathrm{n}(\mathrm{n}>1\) ) different things taken not more than \(\mathrm{r}\) at a time, when each things may be repeated any number of times is (a) \(\left[\left\\{\mathrm{n}\left(\mathrm{n}^{\mathrm{n}}-1\right)\right\\} /\\{\mathrm{n}-1\\}\right]\) (b) \(\left[\left\\{\left(\mathrm{n}^{\mathrm{r}}-1\right)\right\\} /\\{\mathrm{n}-1\\}\right]\) (c) \(\left[\left\\{n\left(n^{r}-1\right)\right\\} /\\{n-1\\}\right]\) (d) \([\\{n(n-r)\\} /\\{n-1\\}]\)
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