Problem
COM-B2-M03-P009 Lower Bounds
#9
★★★☆☆ Level 3 of 5
How many integer solutions does \(x_1+x_2+x_3+x_4=30\) have if \(x_1\ge2\), \(x_2\ge4\), \(x_3\ge0\), \(x_4\ge5\)?
Remove lower bounds by changing variables.
Set \(y_1=x_1-2\), \(y_2=x_2-4\), \(y_3=x_3\), \(y_4=x_4-5\). Then \(y_i\ge0\), and the sum is \(30-2-4-5=19\). The number of nonnegative solutions to \(y_1+\cdots+y_4=19\) is \(\binom{19+4-1}{3}=\binom{22}{3}=1540\).
This is the same stars-and-bars bijection after shifting.