There are 6 Republican, 5 Democrat, and 4 lndependent candidates. How many different ways can a committee of 3 Republicans, 2 Democrats, and 1 Independent be selected?
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
How many different ways can a committee of 3 republicans, 2 democrats and 1 independent be selected
![**Problem Statement:**
*Combinatorial Selection Problem*
There are 6 Republican, 5 Democrat, and 4 Independent candidates. How many different ways can a committee of 3 Republicans, 2 Democrats, and 1 Independent be selected?
**Explanation:**
This problem deals with selecting a specific number of candidates from different groups. It can be solved using combinations, denoted as \( C(n, k) \) or \( \binom{n}{k} \), which represents the number of ways to choose \( k \) items from \( n \) items without regard to order.
1. **Number of Ways to Choose 3 Republicans from 6:**
\[ C(6, 3) = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
2. **Number of Ways to Choose 2 Democrats from 5:**
\[ C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]
3. **Number of Ways to Choose 1 Independent from 4:**
\[ C(4, 1) = \frac{4!}{1!(4-1)!} = \frac{4}{1} = 4 \]
**Total Number of Ways to Form the Committee:**
To find the total number of ways to form the committee, multiply the number of ways to select each group:
\[ 20 \times 10 \times 4 = 800 \]
Therefore, there are 800 different ways to select a committee of 3 Republicans, 2 Democrats, and 1 Independent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd2c3ac26-d64b-48f1-9c9b-c26d07afab9f%2F28eda3b2-32c0-4cc0-9ad7-df3e2a2c1aed%2Fr690sjd.jpeg&w=3840&q=75)
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