The number of positive integral solutions of the inequality 3x + y + z ≤ 30, is

Let w be a non-negative integer such that3x + y + z + w = 30Let a = x – 1, b = y – 1, c = z – 1, d = w, then3a + b + c + d = 25, where a, b, c, d ≥ 0              (1)Clearly, 0 ≤ a ≤ 8. If a = k, thenb + c + d = 25 – 3k                                               (2)Number of non-negative integral solutions of equation (2)=n+r−1Cr=3+25−3k−1C25−3k=27−3kC25−3k=27−3kC2=(27−3k)(26−3k)2=323k2−53k−234∴  Required number =32∑k=08 3k2−53k+234=323⋅8×9×176−538×92+234×9=1215.