How many combinations of towers is possible if you have 2 red blocks 2 black blocks and 3 green blocks ?
Answers
Answer:
If we have a combination of 2 red blocks 2 black blocks and 3 green blocks then we can can have at least 3 tower
Answer:
You have three Orange, three Blue and three Green blocks. In how many different ways can you build a tower of four blocks?
Just in case the blocks are distinct, we can build a tower of 4 blocks in 9P4=3024 ways
Just in case the blocks of the same colour are identical, then we can build the tower by taking
3 blocks of one colour and 1 block of a different colour
2 blocks each of 2 different colours
2 blocks of one colour and 1 block each of distinct colours
Case 1: 3 blocks of one colour and 1 block of a different colour
Three blocks of one colour can be selected in 3 ways. Either they are the oranges or the blues or the greens. Once this is done, we have 3 blocks each of 2 colours remaining. Hence, we can choose the remaining block in 2 ways. After making the selection, we can arrange the blocks in 4!3! ways.
∴ The total number of arrangements
=3×2×4!3!
=3×2×4
=24
Case 2: 2 blocks each of 2 different colours
The two colours to select the the blocks from can be selected in 3C2 ways. Once the colours are selected, we can select the blocks in only 1 way. Having selected, the blocks, we can arrange them in 4!2!2! ways
∴ The total number of arrangements
=3C2×4!2!2!
=3×6
=18
Case 3: 2 blocks of one colour and 1 block each of distinct colours
The colour to be repeated can be chosen in 3 ways. Having selected this colour, we can select 2 blocks of distinct colours in 1 way. Having made the selection, we can arrange them in 4!2! ways.
∴ The total number of arrangements
=3×4!2!
=3×12
=36
∴ The number of different ways of building the towers
=24+18+36
=78