Question

The number of ways in which 10 balls can be selected from 10 identical green balls, 10 identical blue balls and 9 identical red balls is

• A. 63
• B. 64
• C. 65
• D. 66

Answer 1

Answer: (C)

65

Answer 2

Required ways = Coefficient of x¹⁰ in (x⁰ + x¹ + x² .... x¹⁰)² (x⁰ + x¹ + x² + ...+x⁹)

= Coefficient of x¹⁰ in $\left( \frac{1}{1 - x}(1 - x^{11}) \right)^{2}\mspace{6mu}\frac{1}{1 - x}(1 - x^{10})$

= Coefficient of x¹⁰ in (1- x)^-3 (1-x¹⁰) (neglecting the higher power)

= Coefficient of x¹⁰ in (1 – x)^-3 – coefficient of x⁰ in (1 – x)^-3

= $10 + 3 - 1C_{10} - 1 =^{12}C_{10} - 11$

= $\frac{12x11}{2}\mspace{6mu} - \mspace{6mu} 1\mspace{6mu} = \mspace{6mu} 65$