Consider the numbers 1, 2, ..., 100 on the blackboard. Consider the following ru

Question

Consider the numbers 1, 2, ..., 100 on the blackboard. Consider the following rule: We select two numbers, say, x, y from the ones written on the blackboard, we delete them and add the number r y on the blackboard If we apply the rule 99 times then we will be left with only one number on the blackboard. What are all the possible numbers we might end up with? If we apply the rule 98 times then we will be left with two numbers on the blackboard. What are all the possible pairs of numbers we might end up with?

Explanation / Answer

(A) If we apply the rule 99 times then we will left with only one number which is the multiplication of all the numbers from 1 to 100. This is represented by factorial of 100 i.e. 100!

(B)

Consider a situation where we have selected all the numbers till 99 in first 98 selections and we left with 99! and 100.

Similarly, there can be any number from 1 to 100 can be left with the product of the other 99 numbers. Hence there are total 100 possible pairs of numbers.

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