Given there are 5 apples, 10 mangoes and 13 oranges.

Let x₁ is for apple, x₂ is for mango and x₃ is for orange.

Now, first we have to select total 15 fruits out of them.

     x₁ + x₂ + x₃ = 15    (where 0 <= x₁ <= 5, 0 <= x₂ <= 10, 0 <= x₃ <= 13)

=  (x⁰ + x¹ + x² +.........+ x⁵ )*(x⁰ + x¹ + x² +.........+ x¹⁰ )*(x⁰ + x¹ + x² +.........+ x¹³ ) 

= {(1- x⁶ )/(1 - x)}*{(1- x¹¹ )/(1 - x)}*{(1- x¹⁴ )/(1 - x)}

= {(1- x⁶ )*(1- x¹¹ )*{(1- x¹⁴ )}/(1 - x)³

= {(1- x⁶ )*(1- x¹¹ )*{(1- x¹⁴ )} * ∑ ^3-r+1C_r * x^r

= {(1- x¹¹ - x⁶ + x¹⁷ )*{(1- x¹⁴ )} * ∑ ^3+r-1C_r * x^r

= {(1- x¹¹ - x⁶ + x¹⁷ - x¹⁴ + x²⁵  + x²⁰ - x³¹  )} * ∑ ^2+rC_r * x^r

= 1* ∑ ^2+rC_r * x^r - x¹¹ * ∑ ^2+rC_r * x^r - x⁶ * ∑ ^2+rC_r * x^r + x¹⁷ * ∑ ^2+rC_r * x^r - x¹⁴ * ∑ ^2+rC_r * x^r+ x²⁵ * ∑ ^2+rC_r * x^r + x²⁰ * ∑ ^2+rC_r * x^r- x³¹ * ∑ ^2+rC_r * x^r

= ∑ ^2+rC_r * x^r - ∑ ^2+rC_r * x^r+11 - ∑ ^2+rC_r * x^r+6 + ∑ ^2+rC_r * x^r+17 - ∑ ^2+rC_r * x^r+14 + ∑ ^2+rC_r * x^r+25 + ∑ ^2+rC_r * x^r+20 - ∑ ^2+rC_r * x^r+25 

Now we have to find coefficeien of x¹⁵

=  ²⁺¹⁵C₁₅ -  ²⁺⁴C₄  -  ²⁺⁹C₉  -  ²⁺¹C₁   (rest all terms have greater than x¹⁵ , so its coefficients are 0  )

=  ¹⁷C₁₅ -  ⁶C₄  -  ¹¹C₉  -  ³C₁

= ¹⁷C₂ -  ⁶C₂  -  ¹¹C₂  -  ³C₁

= {(17*16)/2} - {(6*5)/2} - {(11*10)/2} - 3 

= (17*8) - (3*5) - (11*5) - 3

= 136 - 15 - 55 - 3

= 136 - 73

= 63

Again we have to distribute 15 fruits between 2 persons.

So  x₁ + x₂ = 15

=  ^2-1+15C₁₅    

= ¹⁶C₁₅     

= ¹⁶C₁   

= 16

Now total number of ways of distriburtion = 16*63 = 1008