A decorator needs to select exactly 30 identical-looking balloons from three available colours: Red, Blue, and Green. To meet the design requirements, they must include at least 4 Red balloons, at least 5 Blue balloons, and at least 6 Green balloons. The total number of different ways the decorator can choose the combination of balloons is

Step-by-step solution

Let x , y , and z be the number of Red, Blue, and Green balloons selected, respectively. According to the given conditions, we have: x + y + z = 30 with x 4 , y 5 , and z 6 . Let x' = x - 4 , y' = y - 5 , and z' = z - 6 . Since x 4 , y 5 , and z 6 , it follows that x' 0 , y' 0 , and z' 0 . Substituting these into the original equation gives: (x' + 4) + (y' + 5) + (z' + 6) = 30 x' + y' + z' = 15 The number of non-negative integral solutions to this equation is given by ^ n+r-1 C_ r-1 , where n = 15 and r = 3 . Numbe