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Chapter 5: Problem 414

At an election 3 wards of a town are canvassed by 4,5 and 8 men respectively. If there are 20 volunteers then the number of ways they can be allotted to different wards is ? (a) \({ }^{20} \mathrm{P}_{4} \cdot{ }^{20} \mathrm{P}_{5} \cdot{ }^{20} \mathrm{P}_{8}\) (b) \({ }^{20} \mathrm{C}_{4} \cdot{ }^{20} \mathrm{C}_{5} \cdot{ }^{20} \mathrm{C}_{8}\) (c) \({ }^{20} \mathrm{C}_{4} \cdot{ }^{16} \mathrm{C}_{5} \cdot{ }^{11} \mathrm{C}_{8}\) (d) \((1 / 3 !){ }^{20} \mathrm{C}_{4} \cdot{ }^{16} \mathrm{C}_{5} \cdot{ }^{11} \mathrm{C}_{8}\)

Short Answer

Expert verified
The short answer is: (c) \({ }^{20} \mathrm{C}_{4} \cdot{ }^{16} \mathrm{C}_{5} \cdot{ }^{11} \mathrm{C}_{8}\).

Step by step solution

01

First, we'll distribute the 20 volunteers among the 3 wards. The first ward has 4 men, the second ward has 5 men, and the third ward has 8 men. So, we need to find the number of ways to allot 4 volunteers to the first ward, 5 volunteers to the second ward, and 8 volunteers to the third ward. #Step 2: Calculate the combinations for each ward#

To find the number of ways to allot volunteers to each ward, we will use the combination formula: \( _{n}C_{r} = \frac{n!}{r! \cdot (n - r)!} \) For the first ward (4 men), we have: \( _{20}C_{4} = \frac{20!}{4!\cdot (20 - 4)!} \) For the second ward (5 men), we must consider that we have already assigned 4 volunteers to the first ward, so there are 16 volunteers left: \( _{16}C_{5} = \frac{16!}{5!\cdot (16 - 5)!} \) For the third ward (8 men), we must consider that we have already assigned 4 volunteers to the first ward and 5 volunteers to the second ward, so there are 11 volunteers left: \( _{11}C_{8} = \frac{11!}{8!\cdot (11 - 8)!} \) #Step 3: Find the total number of ways to allot the volunteers#
02

To find the total number of ways to allot the volunteers, we'll multiply the combinations for each ward: \( _{20}C_{4} \cdot _{16}C_{5} \cdot _{11}C_{8} \) #Step 4: Choose the correct answer from the options given#

The correct answer is (c) \({ }^{20} \mathrm{C}_{4} \cdot{ }^{16} \mathrm{C}_{5} \cdot{ }^{11} \mathrm{C}_{8}\)

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Most popular questions from this chapter

The number of integer \(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\), such that \(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}=20\) and \(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \geq 0\) is (a) \({ }^{24} \mathrm{C}_{3}\) (b) \({ }^{25} \mathrm{C}_{3}\) (c) \({ }^{26} \mathrm{C}_{3}\) (d) \({ }^{27} \mathrm{C}_{3}\)

If a polygon has 90 diagonals, the no. of its sides is given by (a) 12 (b) 11 (c) 10 (d) 15

In chess championship 153 games have been played. If a player with every other player plays only once, then the number of players are (a) 17 (b) 51 (c) 18 (d) 35

A class contain 4 boys and \(g\) girls every sunday 5 students including at least 3 boys go for a picnic to doll house, a different group being sent every week. During the picnic the class teacher gives each girl in the group a doll. If the total number of dolls distributed was 85, then value of \(g\) is (a) 15 (b) 12 (c) 8 (d) 5

If \({ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}\) and \({ }^{\mathrm{n}} \mathrm{P}_{\mathrm{r}}={ }^{\mathrm{n}} \mathrm{P}_{\mathrm{r}+1}\), then the value of \(\mathrm{n}\) is (a) 3 (b) 4 (c) 2 (d) 5
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