Question

in how many ways can 12 of 14 people be distributed into 3 teams where 1st team has 3 members,2nd team has 5 members,3rd team has 4 members​

Answer 1

First select 3 members out of the 14 people for 1st team.

No. of ways of selecting 3 out of 14 people $=\,^{14}C_3.$

3 are selected, and now there are 14 - 3 = 11 people remaining, from whom 5 members have to be selected for 2nd team.

No. of ways of selecting 5 out of 11 people $=\,^{11}C_5.$

Okay 5 are selected, and now there are 11 - 5 = 6 people remaining, from whom 4 members have to be selected for 3rd team.

No. of ways of selecting 4 out of 6 people $=\,^6C_4.$

Hence total no. of ways of selecting 12 out of 14 people for the 3 teams,

$=\,^{14}C_3\cdot\,^{11}C_5\cdot\,^6C_4$

$=\dfrac{14\cdot13\cdot12}{3!}\cdot\dfrac{11\cdot10\cdot9\cdot8\cdot7}{5!}\cdot\dfrac{6\cdot5}{2!}$

$=\underline{\underline{2522520}}$