How many solutions does this inequality have?

The number of solutions is $6+5+4+3+2+1=\boxed{21}$

Possible solutions to the inequalities x+y <= 5 , x,y>=0 and x,y take integer values are

                                           x+y = 0

                                           (x,y)-(0,0)

                                              x+y = 1

                                              (0,1) (1,0)

                                               x+y =2

                                              (0,2),(2,0),(1,1)

                                              x+y = 3

                                             (0,3);(3,0)(1,2);(2,1);

                                            x+y = 4

                                           (0,4)(1,3);(2,2);(3,1);(4,0)

                                           x+y = 5

                                          (0,5);(1,4);(2,3);(3,2);(4,1);(5,0) 

                                         Total number of solutions = 21.