Question

There are nine species of Impatiens (balsams) found in laterite plateaus of the northern Western Ghats, each with a distinct colour. If a plateau has exactly 6 species, then the number of possible colour combinations in the plateau is ….. (Answer in integer).

Hint:

To solve problems involving combinations:
1. Use the formula $\binom{n}{r} = \frac{n!}{r!(n-r)!}$.
2. Simplify factorials step-by-step to avoid errors in large computations.

Answer 1

Step 1: Understand the problem. We are selecting 6 species from a total of 9 species, where the order of selection does not matter. This is a problem of combinations. 
Step 2: Apply the combination formula. The number of ways to choose $r$ objects from $n$ objects is given by the combination formula: $$\binom{n}{r} = \frac{n!}{r!(n-r)!}.$$ Here, $n = 9$ (total species) and $r = 6$ (species to be selected). 
Step 3: Calculate $\binom{9}{6}$. $$\binom{9}{6} = \frac{9!}{6!(9-6)!} = \frac{9!}{6! \cdot 3!}.$$ Simplify: $$\binom{9}{6} = \frac{9 \cdot 8 \cdot 7}{3 \cdot 2 \cdot 1} = 84.$$