I have a large collection of buttons, so this didn't make much of a dint in my collection.
I used approximately 200 for my 37cm letter.
Don't use paper to glue the buttons on to because the buttons will be too heavy and the glue will make the paper not sit flat.
UPDATE: The frame I used was 77x51cm. Clearly it doesn't need to be that large but having the negative space does look nice too.
Make sure your cardboard fits into the photo frame and that there is space for the buttons to fit inside the frame.
Trace the cut out letter onto the card with a grey lead pencil.
I placed it in the centre by eye, but you could place the letter wherever you like in the frame.
Place your buttons onto the outline.
Try and make the gaps in between not too large. It's ok to cross the outline a little bit.
This takes quite a long time!
Make sure you are happy with the arrangement
Get your glue and dip the small paintbrush into it.
Pick up one button at a time and apply a think layer of glue to the back before placing it back down in the same place.
Continue until all are stuck down.
When the glue is dry, give it a good shake to make sure all the buttons are stuck on well.
Re-glue some if necessary.
Put it in the frame and voila!