AMC 10 · 2005 · #18
Grade 7 arithmeticAll of David's telephone numbers have the form 555−abc−defg, where a, b, c, d, e, f, and g are distinct digits and in increasing order, and none is either 0 or 1. How many different telephone numbers can David have?
Pick an answer.
AMC 10 2005 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: A telephone number looks like $555\text{-}abc\text{-}defg$, where the seven digits $a,b,c,d,e,f,g$ are all different, are written in increasing order from left to right, and none of them is $0$ or $1$. Count how many such telephone numbers are possible.
Givens: The seven variable digits are $a,b,c,d,e,f,g$ (the leading $555$ is fixed and does not vary); All seven digits are distinct; The seven digits appear in strictly increasing order left to right; No digit is $0$ or $1$; Answer choices: (A) $1$, (B) $2$, (C) $7$, (D) $8$, (E) $9$
Unknowns: The number of different telephone numbers of this form
Understand
Restated: A telephone number looks like $555\text{-}abc\text{-}defg$, where the seven digits $a,b,c,d,e,f,g$ are all different, are written in increasing order from left to right, and none of them is $0$ or $1$. Count how many such telephone numbers are possible.
Givens: The seven variable digits are $a,b,c,d,e,f,g$ (the leading $555$ is fixed and does not vary); All seven digits are distinct; The seven digits appear in strictly increasing order left to right; No digit is $0$ or $1$; Answer choices: (A) $1$, (B) $2$, (C) $7$, (D) $8$, (E) $9$
Plan
Primary tool: #16 Change Focus / Count the Complement
Secondary: #3 Eliminate Possibilities, #2 Make a Systematic List
The 'increasing order' rule is the hinge: it means arranging the digits is never a choice, so the only decision is which seven digits to use. That turns a scary-looking phone-number count into a plain 'choose a set of digits' count. Tool #3 (Eliminate Possibilities) first trims the digit pool by throwing out $0$ and $1$. Tool #16 (Count the Complement) then does the real work: instead of choosing the seven digits to keep, choose the one digit to leave out — a much smaller decision. Tool #2 (Make a Systematic List) is the fallback that confirms the same total by listing the choices directly.
Execute — Answer: D
4.OA.A.3 Step 1 List the usable digits
- The rule 'none is $0$ or $1$' removes two digits from the ten digits $0$ through $9$.
- What is left is $2,3,4,5,6,7,8,9$.
- Count them: that is $8$ usable digits, and every one of the seven slots must be filled from this pool.
💡 Banning $0$ and $1$ just shrinks the alphabet of digits from ten down to eight.
7.SP.C.8 Step 2 Increasing order fixes the arrangement
- Suppose you have already decided which seven digits to use.
- Because the digits must go in strictly increasing order, there is exactly one legal way to write them: smallest on the left, largest on the right.
- So no two different phone numbers can use the same set of seven digits, and no set of seven digits gives more than one phone number.
- Counting the phone numbers is therefore the same as counting how many ways to pick which seven of the eight digits to use.
💡 Increasing order takes away all freedom in ordering, so only the choice of digits is left to count.
7.SP.C.8 Step 3 Count by leaving one digit out
- Choosing which $7$ of the $8$ digits to keep is the same as choosing which single $1$ digit to leave out — every keep-seven decision matches exactly one leave-out-one decision.
- There are $8$ digits, so there are $8$ different digits you could leave out, giving $8$ possible sets, and therefore $8$ telephone numbers.
- That is choice (D).
💡 Keeping all but one is the same decision as picking the one to throw away, and there are only eight choices for that.
4.OA.A.3 The rule 'none is $0$ or $1$' removes two digits from the ten digits $0$ through 7.SP.C.8 Suppose you have already decided which seven digits to use. Because the digits m 7.SP.C.8 Choosing which $7$ of the $8$ digits to keep is the same as choosing which singl Review
Reasonableness: The answer $8$ sits right inside the answer list and is small, which matches a problem where the arrangement is forced and only a handful of digit-sets exist. A quick sanity list confirms it: leaving out $2$ gives $3456789$, leaving out $3$ gives $2456789$, and so on through leaving out $9$ to give $2345678$ — exactly $8$ numbers, one for each digit dropped. Choice (C) $7$ is the trap for anyone who counts 'seven digits' instead of 'eight ways to choose them,' and (E) $9$ traps anyone who forgets that $1$ is banned and still counts nine digits $1$ through $9$.
Alternative: Count directly with the choose formula instead of the complement. Picking $7$ digits out of $8$ is $\binom{8}{7}=\frac{8!}{7!\,1!}=8$, the same result. Either way the count of ordered arrangements never enters, because the increasing-order rule pins each chosen set to a single phone number.
CCSS standards used (min grade 7)
4.OA.A.3Solve multistep word problems posed with whole numbers (Reading the constraints and reducing the ten digits to the eight usable digits $2$ through $9$.)7.SP.C.8Find probabilities of compound events using organized lists, tables, and simulation (Systematically counting the outcomes: matching each phone number to one chosen set of seven digits, then counting those sets by the complement (which single digit to leave out).)
⭐ When the order is forced, you only count which digits to pick — and keeping $7$ of $8$ digits is the same as choosing the $1$ to drop, so there are $8$ numbers.
⭐ When the order is forced, you only count which digits to pick — and keeping $7$ of $8$ digits is the same as choosing the $1$ to drop, so there are $8$ numbers.
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