Introduction to Card Number Combinations

This article explores the problem of selecting four distinct cards from a set of ten numbered cards, each ranging from 1 to 10, such that their combined sum equals 20. This is a classic example of a combinatorial problem that can be approached through a systematic transformation of variables, ensuring we adhere to constraints and generate valid solutions.

Understanding the Problem

Given a set of 10 cards labeled from 1 to 10, the task is to choose 4 different cards such that their sum equals 20. The challenge lies in ensuring the chosen cards are distinct and their sum meets the specific criterion. Let's walk through the steps to solve this problem.

Transform the Problem

We begin by representing the chosen numbers as (a, b, c, d), where these numbers are distinct and lie within the range from 1 to 10. We aim to find how many such combinations exist.

To simplify the problem, we reframe it using new variables (x_1, x_2, x_3, x_4), where (x_1 a, x_2 b, x_3 c, x_4 d). This transformation maintains the distinctness of the chosen numbers and their sum constraint:

(a b c d 20)

Thus, the transformed equation is:

(x_1 x_2 x_3 x_4 20)

Where (1 leq x_1, x_2, x_3, x_4 leq 10).

Apply Constraints

To ensure the selected numbers are distinct and within the specified range, we can shift the values of the variables as follows:

(y_1 x_1 - 1, y_2 x_2 - 2, y_3 x_3 - 3, y_4 x_4 - 4)

This transformation guarantees that the new variables (y_1, y_2, y_3, y_4) start from 0:

(y_1 1 y_2 2 y_3 3 y_4 4 20)

Which simplifies to:

(y_1 y_2 y_3 y_4 10)

with the constraints (0 leq y_1, y_2, y_3, y_4 leq 6).

Count the Solutions

The task now is to count the number of ways to select four distinct (y_1, y_2, y_3, y_4) from the set {0, 1, 2, 3, 4, 5, 6} (which has 7 elements) such that their sum equals 10. This involves generating all possible combinations of four distinct numbers that meet the sum criterion and checking their validity.

Generate Combinations

To find all valid combinations, we need to iterate through all possible sets of four distinct numbers from {1, 2, ..., 10} and check their sum. The following are the sets of four numbers that sum to 20:

(1, 2, 7, 10) (1, 3, 6, 10) (1, 3, 7, 9) (1, 4, 5, 10) (1, 4, 6, 9) (1, 4, 7, 8) (2, 3, 5, 10) (2, 3, 6, 9) (2, 3, 7, 8) (2, 4, 5, 9) (2, 4, 6, 8) (3, 4, 5, 8) (3, 4, 6, 7)

By carefully counting these combinations, we find that there are 13 valid sets of four numbers.

Thus, the final answer is that there are 13 ways to choose 4 cards such that their sum equals 20.