How many ordered tuples of 7 integers x1,x2,x3,x4,x5,x6,x7 are there, such that
x1+x2+x3+x4+x5+x6+x7−x1x2x3x4x5x6x7=6,
and 1≤xi≤8.
To find the number of ordered tuples of 7 integers that satisfy the given conditions, we can use a technique called recursion.
First, let's denote the number of ordered tuples of length n that satisfy the equation as T(n). We need to find T(7).
To start, let's consider the base case T(1), which represents tuples of length 1. Since 1 ≤ xi ≤ 8, there are 8 possible values for xi. Therefore, T(1) = 8.
Next, we can consider the recursive case. To find T(n), we need to consider the value of the last element, xn. Since 1 ≤ xi ≤ 8, the maximum value that xn can take is 8.
Now, let's consider the equation x1 + x2 + x3 + x4 + x5 + x6 + x7 - x1x2x3x4x5x6x7 = 6. We can rearrange it as follows:
x1 + x2 + x3 + x4 + x5 + x6 + x7 = 6 + x1x2x3x4x5x6x7.
Since 1 ≤ xi ≤ 8, the sum of the seven variables on the left side of the equation can range from 7 (if each variable is 1) to 56 (if each variable is 8). Therefore, the sum on the left side can take 50 different values.
Now, considering the right side of the equation, x1x2x3x4x5x6x7 can take values from 1 to 8^7 (directly related to the maximum value 8 for each xi). However, because the sum on the left side ranges from 7 to 56, we only need to consider the values of x1x2x3x4x5x6x7 that satisfy the condition 6 + x1x2x3x4x5x6x7 = sum.
For each possible value of xn, we can calculate the sum using a loop from 7 to 56, and check if there is a corresponding value of x1x2x3x4x5x6x7 that satisfies the condition.
By iterating through all possible values of xn and summing the total number of valid tuples for each xn, we can calculate T(n) recursively. In this case, we need to calculate T(7).
The final result, T(7), will give us the number of ordered tuples of 7 integers that satisfy the given equation and constraints.