Question

The number of integers greater than 6000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is
( a ) 216
( b ) 192
( c ) 120
( d ) 72

Answer 1

Hint: Firstly, we will find out the number of 4 – digit numbers greater than 6000 and then we will find the number of 5 – digit numbers greater than 6000, then finally we will add them to find total number greater than 600 formed by digits 3, 5, 6, 7 and 8.

Complete step-by-step answer:
Number of integers greater than 6000 will be the number of four digit integers + number of 5 digit integers as we have only five digits provided in the question.
Number of 4 - digit integers which has to be greater than 6000, so the 4 – digit number will start with numbers 6, 7 or 8 not 3 and 5.
So number of ways to select number for thousands place value = 3
Number of ways to select number for hundreds place value = 4
 Number of ways to select number for tens place value = 3
Number of ways to select number for ones place value = 2
So, number of 4 – digit number greater than 6000 = $3\times 4\times 3\times 2$ that is 72……..( i )
For, 5 – digit number we have no restrictions as every 5 – digit number will be greater than 6000.
So, number of ways to select number for ten thousands place value = 5
number of ways to select number for thousands place value = 4
Number of ways to select number for hundreds place value = 3
 Number of ways to select number for tens place value = 2
Number of ways to select number for ones place value = 1
So, number of 5 – digit number greater than 6000 = \[5!=5\times 4\times 3\times 2\times 1\] that is 120……..( ii )
So, total numbers greater than 6000 = 72 + 120 = 192

Note: While solving the questions, try to find all outcome situations which satisfy the condition in question as if any one situation is left, the answer will get wrong. Always remember that, \[n!=n\times (n-1)\times (n-2)\times .......2\times 1\]. Calculation should be done without any error.