AMC 10 · 2017 · #8
Easy mode Grade 4At a party of 30 people, 20 of them all know each other, and the other 10 know no one. Two people who know each other greet with a hug. Two people who do not know each other greet with a handshake. How many handshakes happen in all?
Pick an answer.
AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: At a gathering of $30$ people, $20$ of them all know each other and the other $10$ know no one. Two people who know each other hug; two people who do not know each other shake hands. Find how many handshakes happen in the whole group.
Givens: There are $30$ people in total; $20$ people all know each other (every pair among them knows each other); The other $10$ people know no one (not each other, not the $20$); Acquainted pairs hug; unacquainted pairs shake hands; Answer choices: (A) $240$, (B) $245$, (C) $290$, (D) $480$, (E) $490$
Unknowns: The number of handshakes that occur in the group
Understand
Restated: At a gathering of $30$ people, $20$ of them all know each other and the other $10$ know no one. Two people who know each other hug; two people who do not know each other shake hands. Find how many handshakes happen in the whole group.
Givens: There are $30$ people in total; $20$ people all know each other (every pair among them knows each other); The other $10$ people know no one (not each other, not the $20$); Acquainted pairs hug; unacquainted pairs shake hands; Answer choices: (A) $240$, (B) $245$, (C) $290$, (D) $480$, (E) $490$
Plan
Primary tool: #16 Change Focus / Count the Complement
Secondary: #2 Make a Systematic List, #7 Identify Subproblems
Counting handshakes directly is fiddly, because strangers shake hands across two different groups. But every pair of people does exactly one thing: hug or shake hands. So Tool #16 (Count the Complement) says handshakes = all pairs minus the hugs. Tool #2 (Make a Systematic List) gives the clean way to count pairs: each person meets every other person, then halve to undo the double count. Tool #7 (Identify Subproblems) splits the job into two easy counts — total pairs, then hugging pairs — before one subtraction finishes it.
Execute — Answer: B
4.OA.A.3 Step 1 Hug or handshake, never both
- Pick any two people in the room.
- They either know each other (a hug) or they do not (a handshake) — exactly one of the two.
- So the handshakes are just every pair that is left after the hugs are removed.
- That means: handshakes = (all pairs of people) minus (all hugging pairs).
💡 If each pair does exactly one of two things, counting one kind means subtracting it from the total.
4.NBT.B.5 Step 2 Count every pair in the room
- Count all the pairs among the $30$ people.
- Each person can pair with the other $29$ people, giving $30 \times 29$.
- But that counts each pair twice — once from each person's side — so divide by $2$.
💡 Every person shakes or hugs $29$ others, but you and I share one pair, so halve the total.
4.NBT.B.6 Step 3 Count the hugs
- Hugs only happen between people who know each other, and that is only the $20$-person group.
- Count the pairs inside that group the same way: each of the $20$ pairs with the other $19$, then halve to undo the double count.
💡 The same pair-counting trick works on any group, so the hugs are just the pairs inside the $20$.
3.NBT.A.2 Step 4 Subtract to get the handshakes
- Now remove the hugs from all the pairs.
- The leftover pairs are exactly the handshakes: $435 - 190 = 245$.
- That is choice (B).
💡 Whatever pairs are not hugs must be handshakes, so one subtraction finishes the count.
4.OA.A.3 Pick any two people in the room. They either know each other (a hug) or they do 4.NBT.B.5 Count all the pairs among the $30$ people. Each person can pair with the other $ 4.NBT.B.6 Hugs only happen between people who know each other, and that is only the $20$-p 3.NBT.A.2 Now remove the hugs from all the pairs. The leftover pairs are exactly the hands Review
Reasonableness: The two counts rebuild the whole room: $190$ hugs $+ 245$ handshakes $= 435$, which is exactly the total number of pairs, so nothing is missed or double-counted. The handshake total $245$ sits below $435$ (all pairs) and above $190$ (the hugs), right where it should be.
Alternative: Count handshakes directly. A handshake needs at least one stranger. Strangers-to-acquaintances: each of the $10$ strangers shakes hands with all $20$ acquaintances, giving $10 \times 20 = 200$. Stranger-to-stranger: pairs inside the $10$, which is $\frac{10 \times 9}{2} = 45$. Total $200 + 45 = 245$, matching (B).
CCSS standards used (min grade 4)
4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Setting up the complement plan: handshakes equal the total pairs minus the hugging pairs.)4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number (Multiplying $30 \times 29$ and $20 \times 19$ to count pairs before halving.)4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends (Dividing $870$ and $380$ by $2$ to remove the double-count and get $435$ and $190$ pairs.)3.NBT.A.2Fluently add and subtract within 1000 (Subtracting the $190$ hugs from the $435$ total pairs to get $245$ handshakes.)
⭐ When every pair does exactly one of two things, count all the pairs and subtract the kind you do not want.
⭐ When every pair does exactly one of two things, count all the pairs and subtract the kind you do not want.
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