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:
[tex]x_1+x_2+x_3+x_4+x_5 = 35[/tex]
[tex]x_1 > 0 ; \ \ \ x_1 \ \varepsilon \ Z[/tex]
Thus, the number of the non-negative integral solution is:
[tex]^{(35+3-1)}C_{5-1} = ^{39}C_4[/tex]
(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.
[tex]= \dfrac{(35+4)!}{4!}[/tex]
[tex]= \dfrac{39!}{4!}[/tex]
(c)
Here; provided that the coins are the same and each grandchild gets the same.
Then;
[tex]x_1+x_2+x_3+x_4+x_5 = 35[/tex]
[tex]x_1 > 0 ; \ \ \ x_1 \ \varepsilon \ Z[/tex]
[tex]x_1=x_2=x_3=x_4=x_5[/tex]
[tex]5x_1 = 35\\\\ x_1= \dfrac{35}{5} \\ \\ x_1= 7[/tex]
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 = [tex]\dfrac{35!}{(7!)^55!}[/tex]
Now, distributing them, we have:
[tex]\mathbf{\dfrac{35!}{(7!)^55!} \times 5!= \dfrac{35!}{(7!)^5}}[/tex]
