Problem 55 - Algebra and Trigonometry

Question

The U.S. Senate has 100 members. Suppose that it is desired to place each senator on exactly 1 of 7 possible committees The first committee has 22 members, the second has \(13,\) the third has \(10,\) the fourth has \(5,\) the fifth has \(16,\) and the sixth and seventh have 17 apiece. In how many ways can these committees be formed?

Step-by-Step Solution

Answer

\( \frac{100!}{22! \times 13! \times 10! \times 5! \times 16! \times 17! \times 17!} \)

Step 1: Understand the Problem

The problem involves distributing 100 senators into 7 committees with specified numbers of members in each committee.

Step 2: Write Down the Members of Each Committee

Committee 1: 22 membersCommittee 2: 13 membersCommittee 3: 10 membersCommittee 4: 5 membersCommittee 5: 16 membersCommittee 6: 17 membersCommittee 7: 17 members

Step 3: Use the Formula for Combinations

To find the number of ways to distribute these members, use the multinomial coefficient formula:\[ \binom{n}{k_1, k_2, ..., k_m} = \frac{n!}{k_1! k_2! ... k_m!} \]where:- \(n = 100\) (total senators)- \(k_1 = 22\)- \(k_2 = 13\)- \(k_3 = 10\)- \(k_4 = 5\)- \(k_5 = 16\)- \(k_6 = 17\)- \(k_7 = 17\)

Step 4: Substitute the Values

\[ \binom{100}{22, 13, 10, 5, 16, 17, 17} = \frac{100!}{22! \times 13! \times 10! \times 5! \times 16! \times 17! \times 17!} \]

Step 5: Solve the Computation

Calculate the value using the above formula. The result is a very large number and typically would be computed using software or a calculator designed for large integer computations.

Key Concepts

multinomial coefficient

The multinomial coefficient helps determine the number of ways to distribute objects into different groups. It's an extension of the binomial coefficient. The formula for the multinomial coefficient is given by:
\[ \binom{n}{k_1, k_2, \, ..., \, k_m} = \frac{n!}{k_1! \, k_2! \, ... \, k_m!} \]
In our exercise, we are dividing 100 senators into 7 committees with fixed sizes: 22, 13, 10, 5, 16, 17, and 17 members. Here,

• n = 100 (total senators)
• k_1 = 22 (members in the first committee)
• k_2 = 13 (members in the second committee)
• k_3 = 10 (members in the third committee)
• k_4 = 5 (members in the fourth committee)
• k_5 = 16 (members in the fifth committee)
• k_6 = 17 (members in the sixth committee)
• k_7 = 17 (members in the seventh committee)

Plugging into the formula, we get:
\[ \binom{100}{22, 13, 10, 5, 16, 17, 17} = \frac{100!}{22! \, 13! \, 10! \, 5! \, 16! \, 17! \, 17!} \]
This huge number is the answer we seek for how many different ways we can assign the senators to these specific committees.

factorials

Factorials, represented by the symbol \,\text{!}, play a crucial role in permutations and combinations, including our multinomial distribution. The factorial of a number \,n, written as \, n!, is the product of all positive integers up to \, n. For example:

• \(3! = 3 \times 2 \times 1 = 6\)
• \(4! = 4 \times 3 \times 2 \times 1 = 24\)
• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

Factorials grow rapidly with larger numbers. For instance: \(10! = 3,628,800\).
In our problem, the factorials of the total number of senators (100!) and the factorials of each committee size (like \(22!\), \(13!\), etc.) are used in the multinomial coefficient formula. These factorials help account for all possible arrangements of the senators into the specified committee sizes.

distribution of objects

Distribution of objects into groups is a fundamental concept in combinatorics. It involves assigning a set number of objects (people, items, etc.) into distinct groups. This can be done under different conditions, such as:

• Each group must have a specific number of members.
• All groups combined must account for all objects.

In our specific exercise, we need to distribute 100 senators into 7 predetermined committee sizes: 22, 13, 10, 5, 16, 17, and 17 members. Here’s how it works:

• Each senator can only belong to one committee.
• The order within each committee doesn't matter, but the size does.
• This forms a classic problem for the multinomial coefficient.

By calculating the multinomial coefficient, we determine how many unique ways we can group the senators according to the given committee sizes.