A fruit basket has n identical mangoes, n identical oranges & n distinct fruit. what is the number of ways in which n fruits can be selected from the fruit basket?

(Q). A fruit basket has n identical mangoes, n identical oranges & n distinct fruit. what is the number of ways in which n fruits can be selected from the fruit basket?
Solution:-

Hence total number of ways of selection of n fruits from fruit basket = 
= 1×ⁿCₙ + 2×ⁿCₙ₋₁ + 3×ⁿCₙ₋₂ + ………. + (r+1)×ⁿCᵣ + ……… + (n+1)×ⁿC₀
= 1×ⁿC₀ + 2×ⁿC₁ + 3×ⁿC₂ + ………… + (r+1)×ⁿCₙ₋ᵣ + ……… + (n+1)ⁿCₙ