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: Understanding the Concept
The bag has 7 identical balls and 16 other balls, all different. Identical balls count as one choice for each number taken.

Step 2: Set up the sum
If we take j identical balls (\(j = 0, 1, \dots, 7\)), the other \(12 - j\) come from the 16 distinct balls. So the total is:
\[ \sum_{j=0}^{7} {}^{16}C_{12-j} = {}^{16}C_5 + {}^{16}C_6 + \dots + {}^{16}C_{12} \]

Step 3: Evaluate
Values: \(4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820\). Their sum is 62322.
Check the options: \({}^{18}C_6 = 18564\) and \({}^{18}C_8 = 43758\), giving \(18564 + 43758 = 62322\). That matches option (A). The other options give different totals, for example \({}^{16}C_8 + {}^{18}C_9 = 12870 + 48620 = 61490\).

Final Answer:
The number of ways is \({}^{18}C_6 + {}^{18}C_8\), option (A). \[ \boxed{{}^{18}C_6 + {}^{18}C_8} \]