Math / Algebra

QuestionGiven $(a, b, c, d)$ is a set of integers and all of them are greater than 6 . Find the number of solution $\operatorname{set}(\mathrm{s})$ of $a+b+c+d=50$ .

Studdy Solution

STEP 1

1. $(a, b, c, d)$ is a set of integers.
2. Each integer $a, b, c,$ and $d$ is greater than 6.
3. We need to find the number of solutions for the equation $a + b + c + d = 50$ .

STEP 2

1. Transform the problem to account for the condition that each variable is greater than 6.
2. Use the stars and bars method to find the number of non-negative integer solutions.
3. Calculate the number of solutions.

STEP 3

Transform the problem to account for the condition that each variable is greater than 6.
Since each of $a, b, c,$ and $d$ must be greater than 6, we can set:
$a = a' + 7,$ $b = b' + 7,$ $c = c' + 7,$ $d = d' + 7,$
where $a', b', c',$ and $d'$ are non-negative integers.

STEP 4

Substitute these expressions into the original equation:
$(a' + 7) + (b' + 7) + (c' + 7) + (d' + 7) = 50.$
Simplify the equation:
$a' + b' + c' + d' + 28 = 50.$
Subtract 28 from both sides to find:
$a' + b' + c' + d' = 22.$

STEP 5

Use the stars and bars method to find the number of non-negative integer solutions to the equation $a' + b' + c' + d' = 22$ .
The formula for the number of solutions in non-negative integers of the equation $x_1 + x_2 + \cdots + x_k = n$ is given by:
$\binom{n + k - 1}{k - 1}.$
In this case, $n = 22$ and $k = 4$ .
Calculate:
$\binom{22 + 4 - 1}{4 - 1} = \binom{25}{3}.$

STEP 6

Calculate the binomial coefficient:
$\binom{25}{3} = \frac{25 \times 24 \times 23}{3 \times 2 \times 1} = 2300.$
The number of solutions is:
$\boxed{2300}$

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QuestionGiven $(a, b, c, d)$ is a set of integers and all of them are greater than 6 . Find the number of solution $\operatorname{set}(\mathrm{s})$ of $a+b+c+d=50$ .

Studdy Solution

STEP 1

1. $(a, b, c, d)$ is a set of integers.
2. Each integer $a, b, c,$ and $d$ is greater than 6.
3. We need to find the number of solutions for the equation $a + b + c + d = 50$ .

STEP 2

1. Transform the problem to account for the condition that each variable is greater than 6.
2. Use the stars and bars method to find the number of non-negative integer solutions.
3. Calculate the number of solutions.

STEP 3

Transform the problem to account for the condition that each variable is greater than 6.
Since each of $a, b, c,$ and $d$ must be greater than 6, we can set:
$a = a' + 7,$ $b = b' + 7,$ $c = c' + 7,$ $d = d' + 7,$
where $a', b', c',$ and $d'$ are non-negative integers.

STEP 4

Substitute these expressions into the original equation:
$(a' + 7) + (b' + 7) + (c' + 7) + (d' + 7) = 50.$
Simplify the equation:
$a' + b' + c' + d' + 28 = 50.$
Subtract 28 from both sides to find:
$a' + b' + c' + d' = 22.$

STEP 5

STEP 6

Calculate the binomial coefficient:
$\binom{25}{3} = \frac{25 \times 24 \times 23}{3 \times 2 \times 1} = 2300.$
The number of solutions is:
$\boxed{2300}$

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QuestionGiven (a,b,c,d)(a, b, c, d)(a,b,c,d) is a set of integers and all of them are greater than 6 . Find the number of solution set⁡(s)\operatorname{set}(\mathrm{s})set(s) of a+b+c+d=50a+b+c+d=50a+b+c+d=50.

Studdy Solution

STEP 1

1. (a,b,c,d) (a, b, c, d) (a,b,c,d) is a set of integers.2. Each integer a,b,c, a, b, c, a,b,c, and d d d is greater than 6.3. We need to find the number of solutions for the equation a+b+c+d=50 a + b + c + d = 50 a+b+c+d=50.

STEP 2

1. Transform the problem to account for the condition that each variable is greater than 6.2. Use the stars and bars method to find the number of non-negative integer solutions.3. Calculate the number of solutions.

STEP 3

STEP 4

STEP 5

STEP 6

Calculate the binomial coefficient:(253)=25×24×233×2×1=2300. \binom{25}{3} = \frac{25 \times 24 \times 23}{3 \times 2 \times 1} = 2300. (325​)=3×2×125×24×23​=2300. The number of solutions is:2300 \boxed{2300} 2300​

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QuestionGiven $(a, b, c, d)$ is a set of integers and all of them are greater than 6 . Find the number of solution $\operatorname{set}(\mathrm{s})$ of $a+b+c+d=50$ .

1. $(a, b, c, d)$ is a set of integers.
2. Each integer $a, b, c,$ and $d$ is greater than 6.
3. We need to find the number of solutions for the equation $a + b + c + d = 50$ .

1. Transform the problem to account for the condition that each variable is greater than 6.
2. Use the stars and bars method to find the number of non-negative integer solutions.
3. Calculate the number of solutions.

Calculate the binomial coefficient:
$\binom{25}{3} = \frac{25 \times 24 \times 23}{3 \times 2 \times 1} = 2300.$
The number of solutions is:
$\boxed{2300}$