Question

Sol. [Key Concept: List the colour splits of 8 balls with at least 2 of each (from 5 blue, 6 yellow, 4 red) and sum the products of combinations.]

(But in this questions not mentioned that balls are distinct/alike)?

$$\begin{matrix} & \text{~}\text{Blue -}\text{~}5\text{~}\text{Yellow -}\text{~}6\text{~}\text{Red -}\text{~}4 \\ & \text{~}\text{Reqd. No. of ways}\text{~} \\ & \ = \ ^{5}C_{2} \times \ ^{6}C_{2} \times \ ^{4}C_{4} + \ ^{5}C_{2} \times \ ^{6}C_{3} \times \ ^{4}C_{3} + \ ^{5}C_{2} \times \ ^{6}C_{4} \times \ ^{4}C_{2} \\ & \ + \ ^{5}C_{3} \times \ ^{6}C_{2} \times \ ^{4}C_{3} + \ ^{5}C_{3} \times \ ^{6}C_{3} \times \ ^{4}C_{2} + \ ^{5}C_{4} \times \ ^{6}C_{2} \times \ ^{4}C_{2} \\ & \ = 10 \times 15 \times 1 + 10 \times 20 \times 4 + 10 \times 15 \times 6 + \\ & 10 \times 15 \times 4 + 10 \times 20 \times 6 + 5 \times 15 \times 6 \\ & \ = 150 + 800 + 900 + 600 + 1200 + 450 \\ & \ = 4100 \end{matrix}$$