The recurrence equation
$$\,\,\,\,\,\,\,T\left( 1 \right) = 1$$
$$\,\,\,\,\,\,T\left( n \right) = 2T\left( {n - 1} \right) + n,\,n \ge 2$$
evaluates to

Mala has a colouring book in which each English letter is drawn two times. She wants to paint each of these 52 prints with one of $$k$$ colours, such that the colour-pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of $$k$$ that satisfies this requirement?

How many 4 digit even numbers have all 4 digits distinct?

Two girls have picked 10 roses, 15 sunflowers and 14 daffodils. What is the number of ways they can divide the flowers among themselves?