Math Problem Statement
Consider a collection of cards numbered from 1 to 9. All nine cards are arranged in a row so that the odd numbers are in increasing order, while the even numbers can be in any order. (a) How many ways can this be done? (b) How many ways can eight of these cards be selected and then arranged in a row so that the odd numbers are in increasing order?
Solution
The problem in the image involves arranging cards numbered 1 to 9 under specific conditions. Here’s the breakdown of the two sub-parts of the problem:
Problem Description
- Cards numbered from 1 to 9 are to be arranged in a row.
- Odd-numbered cards must be arranged in increasing order from left to right.
- Even-numbered cards can be placed in any order.
Part (a)
You are asked:
- How many ways can this be done for all nine cards?
Part (b)
You are asked:
- How many ways can eight of these cards be selected and arranged in a row, such that the odd-number
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Permutations
Factorials
Formulas
n! (Factorial formula)
Permutations of even-numbered cards: 4! for part (a), and different cases of selecting 8 cards for part (b)
Theorems
Permutation principle
Suitable Grade Level
Grades 9-12