91Ó°ÊÓ

When dealing with problems in statistics that involve selecting a subset from a larger set, we often use a combination formula. This helps us determine the number of ways to choose items where order does not matter. The combination formula is expressed as \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]. Here, \(n\) represents the total number of items, and \(r\) is the number of items to choose.
In the given exercise, we used the combination formula multiple times:

• To find the total number of ways to form a committee of 7 senators out of 100, we calculated \[\binom{100}{7}\].
• To find the number of ways to select 7 Democrats or 7 Republicans, we computed \[\binom{55}{7}\] and \[\binom{45}{7}\] respectively.

The factorial notation (\(!\)) in the formula signifies the product of all positive integers up to a specified number. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Understanding how to use this formula is crucial for solving problems related to combinations effectively.

Probability expresses how likely an event is to occur. It is calculated as the ratio of favorable outcomes to the total possible outcomes. The formula for probability is \[P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\].
In our problem, we determined several probabilities:

• The probability of selecting all Democrats: Calculated by dividing the number of ways to select 7 Democrats (\binom{55}{7}) by the total number of ways to form the committee (\binom{100}{7}). This resulted in a value of approximately 0.000227 or 0.0227%.
• The probability of selecting all Republicans: Found by dividing \binom{45}{7} by \binom{100}{7}, which yielded around 0.002835 or 0.2835%.
• The probability of selecting 3 Democrats and 4 Republicans: This involved finding the product of \binom{55}{3} and \binom{45}{4}, then dividing by \binom{100}{7}. The resulting probability was roughly 0.2440 or 24.40%.

This approach helps in assigning a quantitative measure to how likely certain combinations of selections are, and is a fundamental concept in statistics for analyzing random events.

The binomial coefficient, often denoted as \(\binom{n}{r}\), indicates the number of ways to choose \(r\) items from \(n\) items without regard to order. It plays an essential role in combinatorics and probability theory.
In our example, \(\binom{55}{7}\) and \(\binom{45}{7}\) are binomial coefficients representing the number of ways to choose 7 members from two different groups. Similarly, \(\binom{55}{3}\) and \(\binom{45}{4}\) are used to select 3 members from 55 and 4 members from 45 respectively.
Here’s a quick breakdown:

• Calculating \(\binom{55}{7}\): \(\frac{55!}{7!(55-7)!}\) = 3,634,421.
• Calculating \(\binom{45}{7}\): \(\frac{45!}{7!(45-7)!}\) = 45,379,620.
• Calculating \(\binom{55}{3}\): \(\frac{55!}{3!(55-3)!}\) = 26,235.
• Calculating \(\binom{45}{4}\): \(\frac{45!}{4!(45-4)!}\) = 148,995.

The importance of binomial coefficients lies in their versatility and frequent application in various statistical methods and probability analyses.
Understanding how to interpret and calculate these coefficients is key to mastering many advanced topics in mathematics and statistics.