注意此句话应该是Do you know how many colours there are in the rainbow?
此句话可以把主句和宾语从句分开理解
即Do you know.
How many colours are there in the rainbow?
楼主可以看到宾语从句how many colours there are in the rainbow （陈述句语序）已改为How many colours are there in the rainbow?（疑问句语序）
宾语从句要用陈述句语序,所以是there are