Answer:
Step-by-step explanation:
Given that:
all coins are same;
The same implies that the number of the non-negative integral solution of the equation:
![x_1+x_2+x_3+x_4+x_5 = 35](https://tex.z-dn.net/?f=x_1%2Bx_2%2Bx_3%2Bx_4%2Bx_5%20%3D%2035)
![x_1 > 0 ; \ \ \ x_1 \ \varepsilon \ Z](https://tex.z-dn.net/?f=x_1%20%3E%200%20%3B%20%20%5C%20%5C%20%5C%20x_1%20%5C%20%20%5Cvarepsilon%20%20%5C%20Z)
Thus, the number of the non-negative integral solution is:
![^{(35+3-1)}C_{5-1} = ^{39}C_4](https://tex.z-dn.net/?f=%5E%7B%2835%2B3-1%29%7DC_%7B5-1%7D%20%3D%20%5E%7B39%7DC_4)
(b)
Here all coins are distinct.
So; the number of distribution appears to be an equal number of ways in arranging 35 different objects as well as 5 - 1 - 4 identical objects
i.e.
![= \dfrac{(35+4)!}{4!}](https://tex.z-dn.net/?f=%3D%20%5Cdfrac%7B%2835%2B4%29%21%7D%7B4%21%7D)
![= \dfrac{39!}{4!}](https://tex.z-dn.net/?f=%3D%20%5Cdfrac%7B39%21%7D%7B4%21%7D)
(c)
Here; provided that the coins are the same and each grandchild gets the same.
Then;
![x_1+x_2+x_3+x_4+x_5 = 35](https://tex.z-dn.net/?f=x_1%2Bx_2%2Bx_3%2Bx_4%2Bx_5%20%3D%2035)
![x_1 > 0 ; \ \ \ x_1 \ \varepsilon \ Z](https://tex.z-dn.net/?f=x_1%20%3E%200%20%3B%20%20%5C%20%5C%20%5C%20x_1%20%5C%20%20%5Cvarepsilon%20%20%5C%20Z)
![x_1=x_2=x_3=x_4=x_5](https://tex.z-dn.net/?f=x_1%3Dx_2%3Dx_3%3Dx_4%3Dx_5)
![5x_1 = 35\\\\ x_1= \dfrac{35}{5} \\ \\ x_1= 7](https://tex.z-dn.net/?f=5x_1%20%3D%2035%5C%5C%5C%5C%20x_1%3D%20%5Cdfrac%7B35%7D%7B5%7D%20%5C%5C%20%5C%5C%20%20x_1%3D%207)
Thus, each child will get 7 coins
(d)
Here; we need to divide the 35 coins into 5 groups, this process will be followed by distributing the coin.
The number of ways to group them into 5 groups = ![\dfrac{35!}{(7!)^55!}](https://tex.z-dn.net/?f=%5Cdfrac%7B35%21%7D%7B%287%21%29%5E55%21%7D)
Now, distributing them, we have:
![\mathbf{\dfrac{35!}{(7!)^55!} \times 5!= \dfrac{35!}{(7!)^5}}](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Cdfrac%7B35%21%7D%7B%287%21%29%5E55%21%7D%20%20%5Ctimes%205%21%3D%20%5Cdfrac%7B35%21%7D%7B%287%21%29%5E5%7D%7D)