특쫑 개념완성 중2-2학기 10강 경우의 수

Main Ideas / Lessons

• The video starts the statistics/probability unit in middle school and emphasizes that to do probability, you must first master counting cases.
• Cases problems are solved by treating events using case breakdown.
• Two recurring logical patterns for counting are:
  1. OR (“either A or B”) → count each alternative separately, then add.
  2. AND (“A and B simultaneously”) → count possibilities for each, then multiply.

Methodology / Instruction Lists

1) OR Rule (conjunction meaning “or”)

When a question says “A or B” (events are alternatives):

• Count the number of cases where A happens.
• Count the number of cases where B happens.
• Add the two counts:

[ #(A \text{ or } B)=#(A)+#(B) ]

The instructor stresses that the “or” word (e.g., Korean “또는/혹은” or English “or”) should be interpreted as always add logic.

Example used

• A restaurant offers:
  • 10 types of Korean food
  • 7 types of Chinese food
• You order one dish
• Since “Korean food or Chinese food” is an OR choice:

[ 10+7=17 ]

Note: The transcript later indicates an answer of “16”, but the described OR counting logic is “add the alternatives,” so the numeric conclusion is inconsistent in the subtitles.

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2) AND Rule (simultaneously, “&”)

When a question says “A and B simultaneously”:

• Let (m=#(A)) (number of cases where A happens)
• Let (n=#(B)) (number of cases where B happens)

If both must occur together (indicated by simultaneously, &, etc.):

[ #(A \text{ and } B)=m\times n ]

The instructor notes that keywords like simultaneously and symbols like & trigger multiplication.

Example used (dice)

Two conditions:

• Event a: the die shows a number ≥ 3
• Event b: the die shows a number that is a divisor of 6

Use the AND rule: compute each event’s case count, then multiply, then adjust if needed.

The transcript ends with a “correct answer number 3,” but the setup/counting details are somewhat garbled.

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3) “Make a line / arrange in a row” (permutation by consecutive selection)

If the problem says “line up N people in a single line”:

• Use decreasing choices for consecutive positions:
  • First position: N
  • Second position: N − 1
  • Next: N − 2
  • Continue until the last spot: 1
• Multiply them:

[ # = N \times (N-1)\times (N-2)\times \cdots \times 1 ]

If you choose k people out of n to stand in a line:

[ # = n \times (n-1)\times (n-2)\times \cdots \times (n-k+1) ]

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4) “Make natural numbers” using digit selection (two-digit numbers)

The method distinguishes whether 0 is included among the available digit cards.

To form a two-digit natural number:

• The tens place cannot be 0
• The ones place can be 0

Case split

Case 1: 0 is NOT included

• Tens place: n choices
• Ones place: (n − 1) choices
• Multiply:

[ n(n-1) ]

Case 2: 0 IS included

• Tens place must exclude 0, so effectively:
  • Tens place: (n − 1) choices
• Ones place:
  • n choices (0 is allowed there)

The transcript mapping is messy, but the key rule is: subtract 1 only for the tens place restriction. The two digit positions are still chosen together using multiplication.

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5) Picture/photo seating in a line (teacher at an end)

The problem involves 4 people total:

• Teacher + 3 students

Key idea:

• The teacher must sit at an end (far left or far right).

Solve via two cases and add:

• Case A: teacher is on the far right
• Case B: teacher is on the far left

In each case:

• Arrange the remaining students in the remaining seats (line order for the photo).
• Then add the totals from both end cases.

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6) Representative Method (counting combinations while avoiding overcounting)

The video introduces the representative method:

• Count by selecting roles/positions,
• but if selections can be permuted without changing meaning, divide out duplicates.

A) Different qualifications (ordered roles are distinct)

Example roles: president and vice president

If there are N people:

• President seat: N choices
• Vice president seat: (N − 1) choices
• Multiply:

[ N(N-1) ]

B) Same qualifications (roles are indistinguishable)

Example: selecting two class monitors (monitor #1 and #2 aren’t distinguished)

• Multiplication overcounts by the number of ways to swap the interchangeable selections.
• For 2 identical roles, divide by 2.

General idea:

• If selecting k people for k identical roles, divide by:

[ k! ]

to remove overcounting due to swapping.

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7) Representative Method for line segments and triangles

Line segments from points

• Choose 2 points out of (n): [ C(n,2)=\frac{n(n-1)}{2} ]

The logic includes:

• multiplication to count ordered selections,
• then division to correct duplicates when endpoints are swapped.

Triangles from points

• Choose 3 points out of (n): [ C(n,3)=\frac{n(n-1)(n-2)}{6} ]

Same idea applies:

• division removes duplicates from permutations of triangle vertices.

Example used (triangles in a labeled set)

• Given 6 points (labeled like A–F in the figure),
• Pick 3 points to form a triangle:
  • count combinations so each triangle is counted once,
  • then divide out duplicates from the different orderings of 3 vertices (implicitly (3!=6)).
• The transcript concludes an answer of 5.

Speakers / Sources Featured

• Wanggaseum (왕가슴) — the instructor/lecturer who narrates and teaches the concepts.