Question 15 of 42intermediate🔧 ApplyNumerical3 marks

Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?

Correct Answer

(even_numbers) 48

(total_4_digit_numbers) 120

Exercise: EXERCISE 6.3 | Q: 4 | (Page 15)
For More Understanding

Explanation

This question tests the application of the multiplication principle for permutations without repetition. The context shows similar examples like Exercise 6.1 Q1(ii) where 3-digit numbers are formed without repetition (5×45 \times 4×3=60), and Example 3 which demonstrates finding even numbers by first filling the unit's place with even digits. The same logic applies here for 4-digit numbers.

Solution Steps

  1. Part 1: Total number of 4-digit numbers

    We need to fill 4 vacant places (thousands, hundreds, tens, units) using digits 1, 2, 3, 4, 5 without repetition.

    • Thousands place can be filled in 5 ways
    • Hundreds place can be filled in 4 ways (remaining digits)
    • Tens place can be filled in 3 ways
    • Units place can be filled in 2 ways

    By multiplication principle: 5×45 \times 4 × 3×23 \times 2 = 120

  2. Part 2: Number of even 4-digit numbers

    For a number to be even, the units digit must be even. From digits 1, 2, 3, 4, 5, the even digits are 2 and 4.

    We start filling from the unit's place (as shown in Example 3):

    • Units place: Can be filled by 2 or 4 = 2 ways
    • Thousands place: Can be filled by remaining 4 digits = 4 ways
    • Hundreds place: Can be filled by remaining 3 digits = 3 ways
    • Tens place: Can be filled by remaining 2 digits = 2 ways

    By multiplication principle: 2×42 \times 4 × 3×23 \times 2 = 48

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