Counters are all alike except colour and there are atleast ten of each colour.How many arrangements will have counters of each colour? There are 10 counters to be picked from 7 different colours. Counters are all alike except colour and there are at least ten of each colour. How many arrangements have at least one counter of each colour? * * * Since there are 7 type of colours.So i took three cases (1)4 alike colour counters,6 different colour counters in $\frac{7\times 10!}{4!}$ (2)3 alike colour counters of one type,2 alike colour counter of other type,5 different colour counters in $\frac{7\times 6\times 10!}{3!2!}$ (3)2 alike colour counters of one type,2 alike colour counter of second type,2 alike colour counter of third type,4 different colour counters in $\frac{7\times 6\times 5 \times 10!}{2!2!2!}$ This totals upto $\frac{103\times 7\times 10!}{24}$ but the answer given in my book is $\frac{49\times 10!}{6}$.I dont know where i am wrong.

Type 3, the choice of colors is $7\choose 3$ rather than $7\times 6\times5$