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).
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: \[ \binom{9}{6} = \frac{9 \cdot 8 \cdot 7}{3 \cdot 2 \cdot 1} = 84. \]
The table lists the top 5 nations according to the number of gold medals won in a tournament; also included are the number of silver and the bronze medals won by them. Based only on the data provided in the table, which one of the following statements is INCORRECT?
Eight students (P, Q, R, S, T, U, V, and W) are playing musical chairs. The figure indicates their order of position at the start of the game. They play the game by moving forward in a circle in the clockwise direction.
After the 1st round, the 4th student behind P leaves the game.
After the 2nd round, the 5th student behind Q leaves the game.
After the 3rd round, the 3rd student behind V leaves the game.
After the 4th round, the 4th student behind U leaves the game.
Who all are left in the game after the 4th round?
A stick of length one meter is broken at two locations at distances of \( b_1 \) and \( b_2 \) from the origin (0), as shown in the figure. Note that \( 0<b_1<b_2<1 \). Which one of the following is NOT a necessary condition for forming a triangle using the three pieces?
Note: All lengths are in meter. The figure shown is representative.