If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety?

Solution:

Combination is defined as

ⁿC_r = [n!/r!(n - r)!]

We have to select 6 bottles randomly with two bottles of each variety,

= ¹⁰C₂ × ⁸C₂ × ¹¹C₂

¹⁰C₂ = [1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10] / [(1 × 2)(1 × 2 × 3 × 4 × 5 × 6 × 7 × 8)]

= 90/2

= 45

Similarly, ⁸C₂ = 28 and ¹¹C₂ = 55

= ¹⁰C₂ × ⁸C₂ × ¹¹C₂

= 45 × 28 × 55

= 69300

Therefore, there are 69300 possible ways of selecting 6 bottles with 2 bottles of each variety.

If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety?

Summary:

If 6 bottles are randomly selected, there are 69300 ways to obtain two bottles of each variety.