Question:medium

A four-digit number is formed by using only the digits 1, 2 and 3 such that both 2 and 3 appear at least once. The number of all such four-digit numbers is

Updated On: Jan 15, 2026
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Correct Answer: 50

Solution and Explanation

To determine the count of four-digit numbers composed solely of digits 1, 2, and 3, where both 2 and 3 are present at least once, we follow these steps:

  • Step 1: Determine the total number of possible four-digit numbers:
    • Each of the four positions can be filled with one of three digits (1, 2, or 3).
    • Total combinations: $3^4 = 81$
  • Step 2: Calculate the number of cases where either 2 or 3 is absent:
    • Numbers formed using only 1 and 3 (no 2): $2^4 = 16$
    • Numbers formed using only 1 and 2 (no 3): $2^4 = 16$
    • Numbers formed using only 1 (no 2 and no 3): $1^4 = 1$
    • Applying the Principle of Inclusion-Exclusion to find cases where at least one of 2 or 3 is missing: $16 + 16 - 1 = 31$
  • Step 3: Subtract excluded cases from the total:
    • Numbers that contain both 2 and 3 at least once: $81 - 31 = 50$
  • Step 4: State the final result:
    • The count of four-digit numbers using only digits 1, 2, and 3, with both 2 and 3 appearing at least once, is 50.

Conclusion: The desired quantity is 50.

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In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
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