Question

A bag contains 23 balls, of which 7 are identical. The number of ways of selecting 12 balls from the bag is....

• A. \(^{18}C_6+^{18}C_8\)
• B. \(^{18}C_7+^{19}C_8\)
• C. \(^{16}C_8+^{18}C_{15}\)
• D. \(^{16}C_8+^{18}C_9\)

Hint:

Take j identical balls (0 to 7) and the rest from the 16 distinct ones, then sum.

Answer 1

The Correct Option is A

Step 1: Model
Selections are decided by how many identical balls are taken (0 to 7) and which distinct balls fill the rest.

Step 2: Sum
$\sum_{k=5}^{12} \binom{16}{k} = 4368 + 8008 + 11440 + 12870 + 11440 + 8008 + 4368 + 1820 = 62322$.

Step 3: Match
$\binom{18}{6} + \binom{18}{8} = 18564 + 43758 = 62322$, so option (A).

Final Answer:
Option (A). \[ \boxed{{}^{18}C_6 + {}^{18}C_8} \]