Question

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

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

Step-by-step solution

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#

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) \({ }^{20} \mathrm{C}_{4} \cdot{ }^{16} \mathrm{C}_{5} \cdot{ }^{11} \mathrm{C}_{8}\)

Key concepts

Permutations

Permutations are arrangements of objects in a specific order. While solving problems related to permutations, the order in which objects are arranged matters. When you have a set of objects and want to figure out how many different ways you can arrange them, permutations is the concept to use. This is especially important when you have a situation where different sequences produce different results or meanings. For example, consider a scenario with 3 different colored balls: red, blue, and green. The order matters because red-blue-green is different from blue-red-green.

• Formula: The number of permutations of \( n \) objects taken \( r \) at a time is given by: \[^{n}P_{r} = \frac{n!}{(n - r)!}\]
• Factorial (\(!\)) plays a vital role in permutation calculations as it helps in determining the number of ways objects can be arranged.

However, in the provided exercise, permutations are not applicable because the order in which volunteers are assigned to wards does not matter.

Combinations

Combinations refer to the selection of objects where the order does not matter. In scenarios where you need to choose objects, but the sequence of selection does not change the outcome, combinations are used.In the election exercise, the townspeople need to allot volunteers to wards. The task requires finding the number of ways to choose volunteers for each ward. Since the order of choosing volunteers doesn't affect who goes to which ward, combinations are the right fit for solving this problem.

• Formula: The number of combinations of \( n \) objects taken \( r \) at a time is: \[^{n}C_{r} = \frac{n!}{r!\cdot(n - r)!}\]
• Step-by-step:
  • For the first ward, calculate \(^{20}C_{4}\), considering all volunteers.
  • For the second ward, calculate \(^{16}C_{5}\) after subtracting the first ward's 4 volunteers.
  • Finally, calculate \(^{11}C_{8}\) for the third ward after subtracting both previous ward allocations.

Factorial

Factorials are a foundational concept in combinatorics, relevant for both permutations and combinations. A factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \). This concept simplifies calculations by providing a structured way to compute the number of ways to arrange or select objects.

• Definition: For any positive integer \( n \), the factorial \( n! \) is calculated as:\[ n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \]
• Special cases:
  • \(0! = 1\), because there is exactly one way to arrange zero objects.
• Uses in combinations: Factorials are used in the numerator to express all possible permutations of \( n \) objects and in the denominator to adjust for non-ordered selections.
  • For instance, \(^{20}C_{4} = \frac{20!}{4!\cdot16!}\), where \(16!\) is used to eliminate counts of order within the 16 unchosen volunteers.