The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of th

Q: Why can't we use the stars and bars formula directly?

Stars and bars assumes each variable can be any non-negative integer. Here, we have an upper bound constraint: each digit must be \le 4. This makes the standard formula invalid because it would allow digits like 5, 6, or 7.

Q: How do we know we found all the partitions of 5?

We systematically list the partitions of 5 into at most 7 parts, where each part is \le 3. By starting with the largest possible part (3) and working downwards, we ensure no case is missed.

Q: Why do we divide by the factorials in the permutation formula?

When we have identical items (like the five 1s in the first case), swapping them doesn't create a new number. Dividing by the factorial of the count of each repeated digit removes these redundant permutations.

Question

The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is _______.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

To form a seven-digit integer using the digits $\{1, 2, 3, 4\}$ such that their sum is exactly $12$, we employ the **Excess Sum** method.

If we fill all seven slots with the minimum digit $1$, the sum is $1 	imes 7 = 7$. Since our target sum is $12$, we have an excess of $12 - 7 = 5$ to distribute among the slots.

Because the maximum value for any digit is $4$, the maximum value we can add to any slot is $4 - 1 = 3$. We must partition the integer $5$ into parts, where each part is $\le 3$.

### Exploring the Partitions

We systematically identify the valid partitions of the excess value $5$:

1.  **Partition $3+2$**: The digits are $\{4, 3, 1, 1, 1, 1, 1\}$.
2.  **Partition $3+1+1$**: The digits are $\{4, 2, 2, 1, 1, 1, 1\}$.
3.  **Partition $2+2+1$**: The digits are $\{3, 3, 2, 1, 1, 1, 1\}$.
4.  **Partition $2+1+1+1$**: The digits are $\{3, 2, 2, 2, 1, 1, 1\}$.
5.  **Partition $1+1+1+1+1$**: The digits are $\{2, 2, 2, 2, 2, 1, 1\}$.

### The Art of Permutation

We calculate the number of distinct arrangements for each set using the multinomial coefficient formula:
$$N = \frac{n!}{n_1! n_2! \dots n_k!}$$

*   **Case 1 $\{4, 3, 1, 1, 1, 1, 1\}$**:
    $$\frac{7!}{5!} = 42$$

*   **Case 2 $\{4, 2, 2, 1, 1, 1, 1\}$**:
    $$\frac{7!}{2!4!} = 105$$

*   **Case 3 $\{3, 3, 2, 1, 1, 1, 1\}$**:
    $$\frac{7!}{2!4!} = 105$$

*   **Case 4 $\{3, 2, 2, 2, 1, 1, 1\}$**:
    $$\frac{7!}{3!3!} = 140$$

*   **Case 5 $\{2, 2, 2, 2, 2, 1, 1\}$**:
    $$\frac{7!}{5!2!} = 21$$

### Final Calculation

Summing the arrangements from all valid cases, we obtain the total number of possible integers:

$$42 + 105 + 105 + 140 + 21 = 413$$

The total number of seven-digit integers satisfying the given conditions is **413**.

Animated Solution for Mathematics - Permutations and Combinations: The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is _______.

Enter Numerical Value:

Visualized Solution

Problem Analysis

The Excess Sum Concept

Case 1: ${4, 3, 1, 1, 1, 1, 1}$

Arranging Case 1

Case 2: ${4, 2, 2, 1, 1, 1, 1}$

Arranging Case 2

Case 3: ${3, 3, 2, 1, 1, 1, 1}$

Arranging Case 3

Case 4: ${3, 2, 2, 2, 1, 1, 1}$

Arranging Case 4

Case 5: ${2, 2, 2, 2, 2, 1, 1}$

Arranging Case 5

Final Summation

The Sigma Insight: Combinations and Selection

Analyzing the Setup

Exploring the Partitions

The Art of Permutation

Final Calculation

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