Find the number of ways in which 5 dice can be rolled to get 25.
While solving this question, the way we solve it is $ x_1 + x_2 + x_3 + x_4 + x_5 $ $ = $ 25 or $ 1 <= x_i <= $ 6
So we replace $ x_i $ by $ y_i = 6-x_i $ , Which one is $ x_i = 6-y_i $
replacing $ x_i $ in the equation above, we get it like → $ (6 * 5) – (y_1 + y_2 + y_3 + y_4 + y_5) $ $ = $ 25
$ (y_1 + y_2 + y_3 + y_4 + y_5) $ = $ 5 $
After solving this equation with the solution solution $ (n-r + 1)! / (n! * (r-1)!) $ we get years like → $ 126
Now consider this problem,
The number of non-negative whole solutions such as $ x_1 + x_2 + x_3 = $ 17 or $ x_1> 1, x_2> 2, x_3> $ 3 is ___________________
Solving this, we solve it as → $ y_1 = x_1-2 $ , $ y_2 = x_2 -3 $ , $ y_3 = x_3-4 $
so, $ x_1 = y_1 + 2 $ , $ x_2 = y_2 + $ 3 , $ x_3 = y_3 + 4 $
Now, we substitute this in our original equation to get →
$ y_1 + 2 + y_2 + 3 + y_3 + 4 = $ 17
$ y_1 + y_2 + y_3 = $ 8
and after solving that, we get the years as $ 45
Now I have a $ DOUB $ here in the second problem since when $ x_1> $ 1 we do it like $ x_1 = y_1 + 2 $ but in the first problem, all the dice should have a value $> 0 $ , so why in this case we did not do $ x_i = y_i + 1 $ for all cases?
And elsewhere if the question was like
$ x_1 + x_2 + x_3 = $ 12 , $ 2 <= x <= $ 5 so how to solve this problem by using an entire solution and applying the formula $ (n-r + 1)! / (n! * (r-1)!) $ ?