Question

A man is distributing his coin collection with 35 coins to his five grandchildren. How many ways are there to distribute the coins if:
a. The coins are all the same
b. The coins are all distinct
c. The coins are the same and each grandchild gets the same number of coins.
d. The coins are all distinct and each grandchild gets the same number of coins.

Answer 1

Answer:

a) $82251$

b) $2.91038 \cdot 10^{24}$

c) $1$

d) $3.17746\cdot 10^{21}$

Step-by-step explanation:

a) If the coins are all the same, we have a combination of 35 identical coins in 5 persons.

The amount of combinations is

$\binom{n+r-1}{r-1}=\binom{35+5-1}{5-1}=\binom{39}{4}=\frac{39!}{4!35!} =82251$

being n the number of coins and r the number of grandchildren.

b) If the coins are different, the number of combinations can be calculated as:

$r^n=5^{35}=2.91038 \cdot 10^{24}$

c) If the coins are all the same, and each grandchild gets the same number of coins, there is only one combination for that (everyone gets 35/5=7 coins)

d)  If the coins are all distinct, but each of the grandchildren get the same numer of coins, the expression to calculate this is:

$\frac{n!}{(n/r)!^r} =\frac{35!}{(35/5)!^5}=\frac{35!}{7!^5}=\frac{1.03331\cdot 10^{40}}{3.25202\cdot 10^{18}} =3.17746\cdot 10^{21}$