Permutations & Combinations
Grade None
Question:
<p>Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appear, is:</p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{2}\)</span>(6!)</p>
<p style="display:inline"><span class="math-tex">\(\frac{5}{2}\)</span>(6!)</p>
<p style="display:inline">5<sup>6</sup></p>
<p style="display:inline">6!</p>
Step-by-Step Solution
Key Concept: To form a 6-digit number using all five specified digits, exactly one digit must be repeated, requiring a combination of digit selection and multiset permutations.
<p>Five digits numbers be 1, 3, 5, 7, 9<br />
For the selection of one digit, we have <sup>5</sup>C<sub>1</sub> choice.<br />
And six digits can be arranged in <span class="math-tex">$\frac{6 !}{2 !}$</span> ways.<br />
Hence, total such numbers = <span class="math-tex">$\frac{5.6 !}{2 !}=\frac{5}{2}$</span>6!</p>
Correct Answer: B