# Basics Of Counting - Combination of Strings

Hello so I just need some help determining wether my logic is correct or if it's flawed.

Bob starts with a string Superman and then applies one modification to it where a modification consists of inserting a number at some position. This process can produce a string like Super5man.

1. How many different strings can Bob create starting with Superman?

_s_u_p_e_r_m_a_n_

So 10 choices and 9 positions: 10 x 90 = 90

Suppose Bob wants to apply two modifications to produce a string like 9Super5man or a string like Super95man.

We want to determine the different strings Bob can create. He does this in two steps:

Step 1. Choose the first number (say 5) and insert it in Superman. (It produces Super5man, etc.)

Step 2. Choose the second number (say 9) and insert it in the string obtained after Step 1. (If the string from step 1 was Super5man, then the result can be 9Super5man or Super95man, etc.)

2. To determine the number of different strings Bob can create, we can just apply the Product Rule. Explain why the product 1 rule will result in overcounting. In particular, what strings will be counted more than once?

So here i'm at a bit of a loss. I'm not too sure what consist of overcounting here.

Would this be considered over counting

95superman 59superman??

3. How would you solve the problem? Explain why you think it is correct.

• You have an error in that you are indeed overcounting. Suppose you in the first step choose to insert a $5$ at the front and in the second step choose to insert a $9$ at the second position resulting in 59superman. Compare this to having in the first step chosen to insert a $9$ at the front and in the second step choosing to insert a $5$ in the first position, again resulting in 59superman. For a correct approach, break into cases based on whether both numbers went into the same "slot" between letters, or whether they went into different slots. – JMoravitz Mar 30 '17 at 21:49