Answer the questions based on the following information. There are four numbers a,b,c,d such that a+b+c+d=12
Find the number of integral solutions of the equation, such that `agt-4, bgt-3,cgt-2,dgt-1`.

Options

A. 1330
B. 1331
C. 1690
D. 340

Answer

To find the number of integral solutions for the equation \( a + b + c + d = 12 \) with the constraints \( a > -4 \), \( b > -3 \), \( c > -2 \), and \( d > -1 \), we can follow these steps:

### Step 1: Rewrite the inequalities
We start by rewriting the inequalities to express \( a, b, c, \) and \( d \) in terms of new variables that are non-negative.

- Since \( a > -4 \), we can let \( a = x_1 - 3 \) where \( x_1 \geq 0 \).
- Since \( b > -3 \), we can let \( b = x_2 - 2 \) where \( x_2 \geq 0 \).
- Since \( c > -2 \), we can let \( c = x_3 - 1 \) where \( x_3 \geq 0 \).
- Since \( d > -1 \), we can let \( d = x_4 \) where \( x_4 \geq 0 \).

### Step 2: Substitute into the equation
Now substitute these new variables into the original equation:

\[ (x_1 - 3) + (x_2 - 2) + (x_3 - 1) + x_4 = 12 \]

This simplifies to:

\[ x_1 + x_2 + x_3 + x_4 - 6 = 12 \]

### Step 3: Rearrange the equation
Rearranging gives us:

\[ x_1 + x_2 + x_3 + x_4 = 18 \]

### Step 4: Apply the formula for integral solutions
Now we need to find the number of non-negative integral solutions to the equation \( x_1 + x_2 + x_3 + x_4 = 18 \). The formula for the number of non-negative integral solutions of the equation \( x_1 + x_2 + \ldots + x_r = n \) is given by:

\[ \binom{n + r - 1}{r - 1} \]

Here, \( n = 18 \) and \( r = 4 \) (since we have four variables: \( x_1, x_2, x_3, x_4 \)).

### Step 5: Calculate the number of solutions
Substituting into the formula gives:

\[ \binom{18 + 4 - 1}{4 - 1} = \binom{21}{3} \]

### Step 6: Calculate \( \binom{21}{3} \)
Now we calculate \( \binom{21}{3} \):

\[ \binom{21}{3} = \frac{21!}{3!(21-3)!} = \frac{21 \times 20 \times 19}{3 \times 2 \times 1} = \frac{7980}{6} = 1330 \]

### Final Answer
Therefore, the number of integral solutions to the equation \( a + b + c + d = 12 \) under the given constraints is \( 1330 \).

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