The number of elements in the set S = x , y , z : x , y , z ∈ Z , x + 2 y + 3 z = 42 , x , y , z ≥ 0 equals ________

Step-by-step solution

Given, x + 2 y + 3 z = 42 , x , y , z ≥ 0 Now, taking cases for different value of z we get, z = 0 ; x + 2 y = 42 Now, y can take values 0 , 1 , 2 , . . . . 21 → 22 elements Similarly, we get z = 1 ; x + 2 y = 39 ⇒ 20 elements z = 2 ; x + 2 y = 36 ⇒ 19 elements z = 3 ; x + 2 y = 33 ⇒ 17 elements z = 4 ; x + 2 y = 30 ⇒ 16 elements z = 5 ; x + 2 y = 27 ⇒ 14 elements z = 6 ; x + 2 y = 24 ⇒ 13 elements z = 7 ; x + 2 y = 21 ⇒ 11 elements z = 8 ; x + 2 y = 18 ⇒ 10 elements z = 9 ; x + 2 y = 15 ⇒ 8 elements z = 10 ; x + 2