Determine the number of 10-combinations of the multiset {3.a, 5 -6,7-c, 00 d}.
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.
![**Problem Statement:**
Determine the number of 10-combinations of the multiset \(\{3 \cdot a, 5 \cdot b, 7 \cdot c, \infty \cdot d\}\).
**Explanation:**
In this problem, you are asked to find the number of ways to choose 10 items from a multiset where:
- 'a' can appear at most 3 times,
- 'b' can appear at most 5 times,
- 'c' can appear at most 7 times,
- 'd' can appear any number of times (infinite).
This involves finding the solutions to the equation:
\[ x_1 + x_2 + x_3 + x_4 = 10 \]
where \(0 \leq x_1 \leq 3\), \(0 \leq x_2 \leq 5\), \(0 \leq x_3 \leq 7\), and \(0 \leq x_4\). Use combinatorial techniques or generating functions to find the number of valid combinations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c802546-7264-4efe-b5e9-2909f0268aed%2Ffd5a842e-dcd4-45cf-bd80-56aca5a00a13%2Fp372g2e_processed.jpeg&w=3840&q=75)
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