Question:
Melvin selects six colours at random and inserts them into a computer program designed to give new colours. A new colour can be formed by mixing any number of colours (2 to 6) from the selected six. How many new colours can the program generate?
Answer:
The program can generate 57 new colours.
Explanation:
There are 6 colours in total, lets say from C1 to C6.
We will get a new colour by mixing any number of colours ranging from 2 to 6 out of the total 6.
The possibilities are as follows:
1) 6C2 (selecting any two colours out of 6)
2) 6C3 (selecting any three colours out of 6)
3) 6C4 (selecting any four colours out of 6)
4) 6C5 (selecting any five colours out of 6)
5) 6C6 (selecting all six colours)
We know that, [tex]nCr = \frac{n!}{( (n-r)! \times r! )}[/tex] where n > r.
[tex]6C2 = \frac{6!}{(4!\times2!)} = 15[/tex],
Similarly, we have 6C3=20, 6C4=15, 6C5=6, 6C6=1
Total 15+20+15+6+1 = 57 possibilities.
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