Permutations & Combinations
Grade None
Question:
<p>How many 5-digit numbers can be made from the digits 1, 2, 3, 4 so that all the digits are taken in each number?</p>
<p style="display:inline">60</p>
<p style="display:inline">240</p>
<p style="display:inline">120</p>
<p style="display:inline">300</p>
Step-by-Step Solution
Key Concept: To form a 5-digit number using all four available digits, one digit must repeat exactly twice, requiring the selection of that repeated digit followed by a permutation of the resulting multiset.
<p>Number of ways to select a digit that is repeated = <sup>4</sup>C<sub>1</sub> = 4<br />
Number of ways to arrange 5 digits with 2 digits alike = <span class="math-tex">\(\frac{5 !}{2 !}\)</span> = 60<br />
<span class="math-tex">\(\Rightarrow\)</span> Required number = 4 <span class="math-tex">\(\times\)</span> 60 = 240</p>
Correct Answer: B