Question

The number of 4 digits number which do not contain 4 different digit is (a) 2432 (b) 3616 (c) 4210 (d) 4464

Short answer

The number of 4-digit numbers which do not contain 4 different digits is 4464.

Step-by-step solution

Calculate Total 4-Digit Numbers

Calculate the total number of 4-digit numbers. Since the first digit can be any of the 9 numbers 1 to 9 and the remaining 3 digits can be any of the 10 numbers 0 to 9, there are 9 × 10 × 10 × 10 = 9000 possible 4-digit numbers.

Calculate 4-Digit Numbers with 4 Different Digits

For a 4-digit number with all 4 digits different, the first digit can be any of the 9 numbers 1 to 9. For the second digit, there are 9 remaining options (0 to 9, excluding the first digit). The third digit can be any of the 8 remaining options, and the fourth digit can be any of the 7 remaining options. So, there are 9 × 9 × 8 × 7 = 4536 4-digit numbers with 4 different digits.

Calculate 4-Digit Numbers with at Least One Repeated Digit

Since we have already calculated the total number of 4-digit numbers and the number of 4-digit numbers with 4 different digits, we can now find the number of 4-digit numbers with at least one repeated digit by subtracting these two values: 9000 - 4536 = 4464. So, there are 4464 4-digit numbers with at least one repeated digit.

Choose the Correct Answer

Comparing this result with the given answer choices, we find that the correct answer is (d) 4464.

Key concepts

Permutations

Permutations are a fundamental concept in combinatorics, often used when you need to find the number of ways to arrange a set of items. In permutations, the order of items matters. This means that switching the positions of two items results in a different permutation. For example, if you have a set of three items {A, B, C}, then the number of ways to arrange them is 6 (ABC, ACB, BAC, BCA, CAB, CBA).

When dealing with problems that require permutations, it's essential to consider how many choices are available for each item in a sequence. This is especially important when the problem involves multiple steps, each with a distinct number of available options. In our exercise, finding the number of 4-digit numbers with different digits is a problem of permutations, since we're interested in arranging these digits in a specific way. In permutations, the formula to calculate the number of arrangements for a set of n items is n!, where n! (n-factorial) represents the product of all positive integers up to n.

Counting 4-digit numbers

Counting 4-digit numbers is a specific application of combinatorial principles. To count 4-digit numbers, we generally start by considering how many choices we have for each digit.
In our exercise, there are specific rules:

• The first digit cannot be zero because the number must be a 4-digit number. Therefore, the first digit has 9 options (1 to 9).
• The second, third, and fourth digits can each be any digit from 0 to 9, giving us 10 options for each.

So, you calculate the total number of possible 4-digit numbers as: \( 9 \times 10 \times 10 \times 10 = 9000 \).These steps help us systematically count possibilities while respecting rules about the range and order of digits.

Repetition

In combinatorics, the concept of repetition is vital. Repetition occurs when some items in a set or sequence can appear more than once. Understanding how repetition impacts counting is crucial.
In the context of counting 4-digit numbers, repetition means that we have variations where one or more digits appear multiple times within the number. Notice how this influences the total count of numbers with fewer unique digits.
To find numbers with at least one repeated digit, subtract the count of numbers with all unique digits from the total count of possible numbers. For instance, if you know there are 9000 total 4-digit numbers and 4536 numbers with all unique digits, then the count of numbers with repetition is:\( 9000 - 4536 = 4464 \).This approach helps to simplify the process of accounting for repeated digits efficiently.