Problem
COM-B2-M03-P004 Positive Solutions
#4
★★☆☆☆ Level 2 of 5
How many positive integer solutions does \(x_1+x_2+x_3+x_4=17\) have?
First set \(y_i=x_i-1\).
Since \(x_i\ge1\), set \(y_i=x_i-1\ge0\). Then \(y_1+y_2+y_3+y_4=13\). By stars and bars, the number of solutions is \(\binom{13+4-1}{4-1}=\binom{16}{3}=560\).
Shows the standard shift from positive to nonnegative solutions.