Question

The domain of the function \(f(x) = ^{(10-x)}C_{(x-6)}\) is...

• A. \([6,10]\)
• B. \(\{6,7,8,9,10\}\)
• C. \([6,8]\)
• D. \(\{6,7,8\}\)

Hint:

For nCr we need r a non-negative integer and r not greater than n.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The symbol \(^nC_r\) needs \(n\) and \(r\) to be non-negative integers with \(0 \le r \le n\). Here \(n = 10 - x\) and \(r = x - 6\).

Step 2: Conditions:
\(x - 6 \ge 0 \Rightarrow x \ge 6\).
\(x - 6 \le 10 - x \Rightarrow 2x \le 16 \Rightarrow x \le 8\).
Also \(x\) must be an integer so that \(n\) and \(r\) are integers.

Step 3: Domain:
\(x \in \{6, 7, 8\}\).

Step 4: Why the other options are wrong.
Intervals such as \([6,10]\) or \([6,8]\) include non-integers, which are not allowed. The set \(\{6,7,8,9,10\}\) includes 9 and 10, where \(r > n\) (for \(x = 9\), \(n = 1\) and \(r = 3\)).

Final Answer:
The domain is \(\{6, 7, 8\}\), option (D). \[ \boxed{\{6,7,8\}} \]