Question

There are 3 Mangoes, 5 Oranges and 6 Apples in a fruit basket. Number of ways in which 10 fruits can be selected from the basket is

• A. 67
• B. 38
• C. 30
• D. 29

Answer 1

Answer: (D)

29

Answer 2

Required ways

= coefficient of x¹⁰ in (1+x+x²+x²) (1+x+x²+...+x⁵) (1+x+x²+....+x⁶)

= coefficient of x¹⁰ in

$\frac{1}{1 - x}(1 - x^{4})\frac{1}{1 - x}(1 - x^{6})\frac{1}{1 - x}(1 - x^{7})$ = coefficient of x¹⁰ in (1-x)^-3 (1-x⁴) (1-x⁶) (1-x⁷)

= coefficient of x¹⁰ in (1-x)^-3 (1 – x⁴ – x⁶ – x⁷ + x¹⁰) (neglecting higher power)

= coefficient of x¹⁰ in (1 – x)^-3 coefficient of x⁶ in (1 – x)^-3

- coefficient of x³ in (1 – x)^-3 + coefficient of x⁰ in (1- x)^-3

= $12C_{10} -^{8}C_{6} -^{5}C_{3} +^{3}C_{3}$

= 66 – 28 – 10 + 1= 29