A committee of k members was formed from the n members of a certain company's board of directors. Each member...

Phase 1: Owning the Dataset

Visualization

Let's create a simple diagram to understand the email flow:

Board of Directors (n members total)
┌─────────────────────────────────────┐
│  Committee (k)  │  Non-Committee     │
│                 │  (n - k)           │
│  ● ● ● ... ●   │  ○ ○ ○ ... ○      │
│  k members      │  (n-k) members     │
└─────────────────────────────────────┘
         ↓
  Each ● sends one email to each ○
  Total emails = k × (n - k) = 35

Understanding the Problem

• We have n board members total
• k members form a committee
• Each committee member sends exactly one email to each NON-committee member
• No emails sent within the committee or to anyone else
• Total emails sent = 35

Phase 2: Understanding the Question

Mathematical Relationship

Each committee member sends emails to (n - k) non-committee members.
With k committee members, total emails = \(\mathrm{k} \times (\mathrm{n} - \mathrm{k}) = 35\)

What We're Looking For

We need values of k and n from the given choices that satisfy:

• \(\mathrm{k} \times (\mathrm{n} - \mathrm{k}) = 35\)
• \(\mathrm{k} < \mathrm{n}\) (committee must be smaller than the whole board)
• Both k and n must be from our choices: [4, 5, 6, 10, 12]

Phase 3: Finding the Answer

Systematic Testing

Let's test each possible value of k:

If k = 4:
\(4(\mathrm{n} - 4) = 35\)
\(4\mathrm{n} - 16 = 35\)
\(4\mathrm{n} = 51\)
\(\mathrm{n} = 12.75\) (not an integer, so invalid)

If k = 5:
\(5(\mathrm{n} - 5) = 35\)
\(5\mathrm{n} - 25 = 35\)
\(5\mathrm{n} = 60\)
\(\mathrm{n} = 12\) ✓

Verification: \(5 \times (12 - 5) = 5 \times 7 = 35\) ✓

? Stop here - we found our answer.

Phase 4: Solution

Final Answer

• Statement 1 (k): 5
• Statement 2 (n): 12

The committee has 5 members from a board of 12 members. Each of the 5 committee members sends one email to each of the 7 non-committee members, totaling \(5 \times 7 = 35\) emails.