Question

A child has 20 deciduous teeth. Two of her teeth are decayed. Given that this is all that you currently know about the child’s dentition, choose from the options that the possible combinations of decayed teeth she might have?

• A. 10
• B. 380
• C. 180
• D. 190

Hint:

Always check whether the question asks for combination (order doesn’t matter) or permutation (order matters). Here, it is a combination problem.

Answer 1

The Correct Option is B

Step 1: Problem Definition.
There are 20 deciduous teeth. The task is to determine the number of ways to select any 2 of these teeth for decay, assuming the order of selection does not matter. This is a combination problem.
Step 2: Combination Formula.
The formula to calculate combinations of choosing $r$ items from a set of $n$ items is:
\[^nC_r = \frac{n!}{r!(n-r)!}\] Step 3: Value Application.
Given $n = 20$ (total teeth) and $r = 2$ (teeth to be decayed). Substitute these values into the formula:
\[^{20}C_2 = \frac{20!}{2!(20-2)!} = \frac{20 \times 19}{2} = 190\] Step 4: Consideration of Variations.
While the direct calculation yields $190$, alternative interpretations such as symmetrical arrangements (e.g., upper vs. lower jaw) or permutations might lead to different results in specific problem contexts. However, $190$ represents the fundamental and correct combinatorial solution.
Final Answer:
\[\boxed{190}\]