Permutations & Combinations
Grade None
Question:
<p>How many four digit numbers can be made from the digits 2, 3, 4, 5 and 6 without repeating any digit such that they are exactly divisible by 4?</p>
<p style="display:inline">120</p>
<p style="display:inline">5</p>
<p style="display:inline">6</p>
<p style="display:inline">36</p>
Step-by-Step Solution
Key Concept: A number is divisible by 4 if its last two digits form a multiple of 4, so list all such pairs from the available digits and multiply by the permutations of the remaining digits.
<p>A number is divisible by 4 if the number formed by its last two digits (ten’s and units) is exactly divisible by 4. The possible digits at ten’s and unit’s places can be (- - 24), (- - 32), (- - 36), (- - 52), (- - 56) and ( - - 64) The number of ways to fill ten’s and unit’s places = 6<br />
The thousand’s and hundred’s places can be filled from the remaining 3 digits which can be done in <sup>3</sup>P<sub>2</sub> = 6 ways<br />
<span class="math-tex">$\Rightarrow$</span> Required number four digit numbers<br />
= 6 <span class="math-tex">$\times$</span> 6 = 36</p>
Correct Answer: D