Finding the number of non-negative solutions

Dear students,

Sometimes you might be wondering how to score well, just by solving questions easily by knowing the simple fundamentals. Here goes such few examples to solve complicated problems in a very simple way, just by knowing the trick behind it.

1) The number of non-negative integral solutions of x + 2y + 3z = 101 is

(a) 884 (b) 901 (c) 3434 (d) 2312

2) The number of non-negative integral solutions of 3x + 2y + z = 200 is

(a) 2312 (b) 901 (c) 1037 (d) 3434

3) Number of non-negative integral solutions of x₁ + x₂ + x₃ + x₄ + x₅ = 20 and x₁ + x₂ = 15 is

(a) 37 (b) 336 (c) 20 (d) 84

4) 101 mangoes (Identical) are to be distributed to 3 persons A, B, C so that B gets even number of mangoes and C gets multiples of 3 mangoes. Number of ways of distribution is

(a) 901 (b) 2312 (c) 1037 (d) 3434

5) 200 biscuits (Identical) are to be distributed among 3 boys X, Y, Z so that y gets multiples of 2 and z gets multiples of 3 biscuits. Number of ways of distributing the biscuits is

a) 884 b) 901 c) 3434 d) 2312

6) In how many ways can a dice be thrown thrice by a person to make a sum = 12 which is similar to the problem number of non-negative integral solutions of linear equation x₁ + x₂ + x₃ = 12
7) Find the non-negative integral solutions of 3x + y + z = 24 or x + 2y + 3z = 101 or x + 2y + z = 200 or 3x + 5y + 7z = 54 or x + y + z = 6.

These type of problems can be solved in the following way

Solution: All the above problems can be represented by a general equation a₁x₁ + a₂x₂ + a₃x₃ = n, where a₁, a₂, a₃ are integers which is similar to number of non-negative integral solution of a₁x₁ + a₂x₂ + a₃x₃ = n

Hence, the number of non-negative integral solutions = Coefficient of [xa+ a + a] in [(1 + xa + xa+1 + …) x [(1 + xa + xa+1 + …) x [(1 + xa + xa+1 + …)]

To find the coefficient of [xa+ a + a] is little complicated and it takes more time to solve. Which can be solved simply by the following formula.

$$\left[ =\frac{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+3\left( {{a}_{1}}{{a}_{2}}+{{a}_{2}}{{a}_{3}}+{{a}_{3}}{{a}_{1}} \right)}{12{{a}_{1}}{{a}_{2}}{{a}_{3}}}+\frac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}}{2{{a}_{1}}{{a}_{2}}{{a}_{3}}}n+\frac{1}{2{{a}_{1}}{{a}_{2}}{{a}_{3}}}{{n}^{2}} \right]$$.