Question

In a meeting held by ACS, there were 10 attendees. If N attendees are selected to deliver speeches during the meeting, and it is given that N is a positive integer, what is the value of N?
Statement 1: N is a multiple of 3.
Statement 2: There are 210 ways to select the N attendees to deliver speeches.

Hint:

In combinatorics problems, if the wording is ambiguous ("select to perform a task"), test both combination and permutation formulas against the numbers given in the statements. The interpretation that yields a valid integer solution is almost always the intended one.

Answer 1

Step 1: Understanding the Concept:
This data sufficiency problem involves combinatorics. The key is to correctly interpret the phrase "select the N attendees to deliver speeches". This phrase can be ambiguous. It could mean simply choosing a group (a combination), or choosing a group and arranging them in an order (a permutation). We must test which interpretation makes sense with the given numbers.
Step 2: Analyze the Statements:
Let's first assume the phrase implies a permutation (selecting N people and arranging their speaking order). The number of ways would be \(P(10, N) = \frac{10!}{(10-N)!}\).
Let's test this with Statement 2: \(P(10, N) = 210\).
\(P(10, 1) = 10\)
\(P(10, 2) = 10 \times 9 = 90\)
\(P(10, 3) = 10 \times 9 \times 8 = 720\)
There is no integer N for which \(P(10, N) = 210\). This interpretation is unlikely to be correct.
Let's now assume the phrase implies a combination (simply selecting a group of N speakers, with the order not being important). The number of ways would be \(C(10, N) = \frac{10!}{N!(10-N)!}\).
Let's test this with Statement 2: \(C(10, N) = 210\).
\(C(10, 1) = 10\)
\(C(10, 2) = \frac{10 \times 9}{2} = 45\)
\(C(10, 3) = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120\)
\(C(10, 4) = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 10 \times 3 \times 7 = 210\)
So, \(N=4\) is a solution.
Due to the symmetric property of combinations, \(C(n, k) = C(n, n-k)\), we also have:
\(C(10, 10-4) = C(10, 6) = C(10, 4) = 210\).
So, \(N=6\) is also a solution.
This interpretation leads to valid integer solutions, so it is the most likely intended meaning.
Analysis of Statements based on Combination interpretation:
Statement 1: N is a multiple of 3.
Since \(1 \le N \le 10\), N could be 3, 6, or 9. This statement alone does not give a unique value for N. Therefore, Statement (1) is NOT sufficient.
Statement 2: There are 210 ways to select the N attendees to deliver speeches.
As calculated above, this implies \(C(10, N) = 210\), which means N could be 4 or 6. Since there are two possible values for N, Statement (2) alone is NOT sufficient.
Combining Statements (1) and (2):
From Statement 1, we know N must be 3, 6, or 9. From Statement 2, we know N must be 4 or 6. The only value that satisfies both conditions is \(N=6\). Together, the statements provide a unique value for N. Therefore, BOTH statements TOGETHER are sufficient.
Step 3: Final Answer:
Neither statement is sufficient on its own, but they are sufficient when combined.