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Chapter 2: Problem 40

There are \(n\) sticks lined up in a row, and \(k\) of them are to be chosen. (a) How many choices are there? (b) How many choices are there if no two of the chosen sticks can be consecutive? (c) How many choices are there if there must be at least \(l\) sticks between each pair of chosen sticks?

Short Answer

Expert verified
The solution to the problem: (a) There are \(C(n, k)\) choices. (b) If no two of the chosen sticks can be consecutive, there are \(C(n - k + 1, k)\) choices. (c) If there must be at least \(l\) sticks between each pair of chosen sticks, there are \(C(n - l(k-1), k)\) choices.

Step by step solution

01

Solving Part (a)

For part (a), the problem is a straightforward combination as there are no restrictions. As you're choosing \(k\) out of \(n\) without restrictions, you'll use the combination formula \(C(n, k) =\frac{n!}{k!(n-k)!}\) where \(n\) is the total number of sticks lined up and \(k\) is the number of them to be chosen.
02

Solving Part (b)

In this case, if no two of the chosen sticks can be consecutive, then \(k\) sticks are to be chosen from a total of \(n - k + 1\) sticks, so the calculation is \(C(n - k + 1, k)\).
03

Solving Part (c)

In this case, if there must be at least \(l\) sticks between each pair of chosen sticks, there should be \(l(k-1)\) sticks that cannot be chosen. So, again it's a problem of to choose \(k\) out of \(n - l(k-1)\) sticks, which makes the calculation using formula to be \(C(n - l(k-1), k)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics
Combinatorics is a branch of mathematics concerned with counting, both as a means to understand quantitative data and as a tool for problem-solving. It often involves assessing the different permutations and combinations of sets. In the context of the problem with sticks, combinatorics helps us explore how we choose a certain number of sticks from a lineup.
  • Permutations: In permutations, the order of selection matters. If we were concerned about the sequence in which we pick sticks, permutations would be applicable.
  • Combinations: Our focus is on combinations, where the order does not matter. This means if we pick a stick first or last, it doesn't impact the combination count as long as the same sticks are picked.
Combinatorics allows us to calculate these outcomes efficiently without having to manually count each possible selection.
Combinatorial Restrictions
Combinatorial restrictions are limitations or conditions placed on the selection process in combinatorial problems. In our exercise about picking sticks, we encountered different restrictions in parts (b) and (c).
  • In part (b), we cannot have any two chosen sticks be consecutive, introducing a restriction on adjacent pairs.
  • In part (c), we must leave a minimum of certain sticks (denoted by \(l\)) between each chosen pair, adding another layer of complexity.
These restrictions change the total number of available choices and require adjusted calculations, unlike a straightforward choice without any restrictions.
Combination Formula
The combination formula is a mathematical way to determine the number of ways to choose a subset of items from a larger set without regard to the order in which items are chosen. It is expressed as:\[ C(n, k) = \frac{n!}{k!(n-k)!} \]This formula helps solve the first part of our stick problem, where we choose \(k\) sticks from \(n\) without any restrictions. Here's what the components mean:
  • \(n!\): Factorial of the total number of items from which selections are made.
  • \(k!\): Factorial of the number of selections.
  • \((n-k)!\): Factorial of the remaining unselected items.
Using the combination formula, we calculate the number of possible combinations without considering order and restrictions.
Non-Consecutive Selections
Non-consecutive selections involve choosing items such that no two selected items are adjacent to each other. In our exercise, this condition appeared in part (b). Here's how it's addressed:
  • Imagine the sticks as a series of slots or positions, where a gap is needed between each selected stick.
  • Instead of choosing \(k\) directly from \(n\), this requirement means we are effectively selecting from \(n - k + 1\) options to account for the gaps needed.
  • The calculation thus becomes \(C(n - k + 1, k)\), accounting for these non-consecutive constraints.
This subtle consideration of the gaps fundamentally changes the approach and required adjustments to the original calculation methods.

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