Question

How many different nine digit numbers can be formed from the number \(22,33,55,888\) by rearranging its digits so that the odd digits occupy even positions? (a) 16 (b) 4 (c) 60 (d) 5

Short answer

The total number of different nine-digit numbers that can be formed by rearranging the digits of the number $22,33,55,888$ such that odd digits occupy even positions is 60. The correct answer is (c) 60.

Step-by-step solution

Arrange the Odd Digits

Since there are 4 even positions available, the odd digits can be arranged in 4!/(2!2!) ways. The denominator accounts for the fact that there are 2 threes and 2 fives, which are repetitive. Calculate the total number of ways: 4!/(2!2!) = \(\frac{4!}{2!2!} = \frac{(4)(3)(2)(1)}{(2)(1)(2)(1)} = 6\)

Arrange the Remaining Digits

The remaining digits are two 2s and three 8s to be placed in the odd positions 1, 3, 5, 7, and 9. These can be arranged in 5!/(2!3!) ways. Calculate the total number of ways: 5!/(2!3!) = \(\frac{5!}{2!3!} = \frac{(5)(4)(3)(2)(1)}{(2)(1)(3)(2)(1)} = 10\)

Multiply the Arrangements

Now we have found the number of ways to arrange both the odd and even digits, we need to multiply them to get the total number of ways to arrange the entire nine-digit number. Total arrangements = (Odd digit arrangements) × (Even digit arrangements) = 6 × 10 = 60 Thus, there are 60 different nine-digit numbers that can be formed by rearranging the digits of the number 22,33,55,888 such that odd digits occupy even positions. The correct answer is (c) 60.

Key concepts

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It's akin to solving a puzzle where you figure out the number of ways pieces can be arranged following specific rules. In our exercise, the puzzle is to determine the number of distinct nine-digit numbers constructed from the given digits by rearranging them while ensuring odd digits are in even positions.

This involves understanding permutations (different arrangements from a set of objects where order matters) and combinations (selections of objects where the order does not matter). By focusing on permutations, we address the specific order of digits to satisfy the condition of odd digits occupying even positions.

Factorial

The factorial, denoted by an exclamation mark (!), is a function that multiplies a number by every number below it down to 1. For example, the factorial of 4, written as '4!', is calculated as \( 4 \times 3 \times 2 \times 1 = 24 \). Factorials are fundamental in combinatorics for calculating permutations and combinations. They serve as the building blocks for arranging objects in order, as demonstrated in our problem where we use factorials to find the number of different arrangements possible for the set of given digits.

Arrangement of Numbers

The arrangement of numbers in specific patterns or orders is a common combinatorics problem. In our exercise, the numbers must be rearranged so that odd digits are in the even positions. This creates a restriction which is the key element of such problems. The restriction influences how we calculate permutations because we have to respect the rule while considering all possible arrangements. In simpler terms, think of it as organizing books on a shelf where certain books must only be placed on certain positions, significantly reducing the way the books can be arranged.

Permutation Formula

The permutation formula is used to calculate the number of ways we can arrange a subset of a set. When the order of arrangement is important, the formula is usually \( n! / (n-r)! \), where 'n' is the total number of items to choose from, and 'r' is the number of items to arrange.

However, in the case of our exercise, we have repeated digits, which means we need to modify the formula to \( n! / (p_1! \times p_2! \times \text{...} \times p_k!) \), where each 'p' represents the factorial of the count of identical items. This advanced permutation formula allows us to calculate the unique arrangements of our number accurately, accounting for the repeated digits to avoid overcounting. It is used twice in the exercise: first, to arrange the odd numbers in even positions while accounting for repeated threes and fives, and then to arrange the remaining even digits, accounting for the repeated twos and eights.