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: Test values:
$x = 6$: $^4C_0 = 1$, fine. $x = 7$: $^3C_1 = 3$, fine. $x = 8$: $^2C_2 = 1$, fine. $x = 9$: $^1C_3$, invalid since 3 > 1. $x = 5$: $r = -1$, invalid.

Step 2: Reason:
The valid integers are exactly those with $6 \le x \le 8$.

Step 3: Note:
Because $n$ and $r$ must be integers, non-integer values such as $6.5$ are not in the domain.

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