IPMAT 2025 Exam Solved Papers

We are being given with six digits- 1,3,5,7,8,9So for a number formed by these digits to be greater than 5000, the number can be either 4-digit, 5-digit or 6-digiti.) 4-digit number:Since the number is divisible by 5, so last digit has to be 5Also, the number is greater than 5000 so first digit has to be 7, 8 or 9So, the first digit can be selected in ${ }^{3} C_{1}$ waysFor the remaining 2-places, we can select 2 digits from the remaining 4 digits in ${ }^{4} C_{2}$ ways and arrange them in 2! waysSo, number of ways of forming the 4-digit number ${ }^{3} C_{1} \times{ }^{4} C_{2} \times 2!=36$ waysii.) 5-digit number:The last digit has to be 5 as the number is divisible by 5Now, for the remaining 4 places we have to select 4 digits from 5 digits which can be done in ${ }^{5} C_{4}$ ways and rearrange them in 4! waysSo, number of ways of forming the 5-digit number ${ }^{5} C_{4} \times 4!=120$ waysiii.) 6-digit number:The last digit has to be 5 as the number is divisible by 5So, the remaining 5 places can be filled by remaining 5 digits in 5! = 120 waysSo, number of ways of forming the 6-digit number 5! = 120 waysSo, number of integers = 36+120+120=276