A home owner wants to purchase two different pictures, one to hang above a hall table and one to hang above a sofa. An interior decorator arrives at the home with several different pictures and shows the owner all 42 different arrangements. How many different pictures did the decorator show the home owner.
My work:
If a home owner wants to purchase 2 pictures we need to make our equation so 2!/42= 21
the answer is 21
The interior decorator comes to the home with $N$ pictures, then he shows the pictures to the owner in pairs.
We are looking for the number $N$ of pictures the decorator brought given that there are $42$ arrangements of pairs.
Ok suppose the pictures are shuffled and the decorator takes one of the $N$ pictures, then there are now $N-1$ pictures to take from and so because he needs one more picture to show to the home owner he takes one. Therefore there could be $N(N-1)$ possible choices he could've made. Then the number of pictures amounts to solving $N(N-1)=42$ for some positive integer $N$, can you solve it?