Answer:
210
Step-by-step explanation:
We are given that
Total number of peoples=n=10
Number of different phone lines=r=4
We have to find the number of groups of 4 people can answer 4 lines when all lines begin to ring at once.
Using combination formula
![nC_r=\frac{n!}{r!(n-r)!}](https://tex.z-dn.net/?f=nC_r%3D%5Cfrac%7Bn%21%7D%7Br%21%28n-r%29%21%7D)
Substitute the values in the given formula
Then , we get number of groups of 4 people can answer 4 lines=![10C_4](https://tex.z-dn.net/?f=10C_4)
Number o groups of 4 peoples=![\frac{10!}{4!(10-4!}=\frac{10!}{4!6!}](https://tex.z-dn.net/?f=%5Cfrac%7B10%21%7D%7B4%21%2810-4%21%7D%3D%5Cfrac%7B10%21%7D%7B4%216%21%7D)
Number o groups of 4 peoples=![\frac{10\times 9\times 8\times 7\times 6!}{4\times 3\times 2\times 1\times 6!}=10\times 3\times 7=210](https://tex.z-dn.net/?f=%5Cfrac%7B10%5Ctimes%209%5Ctimes%208%5Ctimes%207%5Ctimes%206%21%7D%7B4%5Ctimes%203%5Ctimes%202%5Ctimes%201%5Ctimes%206%21%7D%3D10%5Ctimes%203%5Ctimes%207%3D210)
Hence, 210 groups of 4 peoples can be made and answer these 4 lines.