permutations - Calculating the probability of receiving all possible rewards after 15 events

I encountered this question in my Data Management and Statistics textbook. I tried to calculate the probability using binomial theorem and combinations/permutations, but I could only get close to the answer. I would really appreciate any help with this question:

If Joe buys a cereal box and it has the following probabilities of giving one of the five possible rewards:

• Toy Car: 20%


• Toy Truck: 30%


• Toy Spoon: 10%


• Toy Doll: 35%


• Toy Gun: 5%

What is the probability of Joe getting all the rewards after buying 15 cereal boxes?

Thank you in advance

Answer

Use the Inclusion/Exclusion principle:

1. Start with $1$




2. Subtract the following:

   
   
   
   
   
   
   • The probability of not winning C is $(1-0.20)^{15}$
   
   
   • The probability of not winning T is $(1-0.30)^{15}$
   
   
   • The probability of not winning S is $(1-0.10)^{15}$
   
   
   • The probability of not winning D is $(1-0.35)^{15}$
   
   
   • The probability of not winning G is $(1-0.05)^{15}$
   
   



3. Add the following:

   
   
   
   
   
   • The probability of not winning C,T is $(1-0.20-0.30)^{15}$
   
   
   
   • The probability of not winning C,S is $(1-0.20-0.10)^{15}$
   
   
   • The probability of not winning C,D is $(1-0.20-0.35)^{15}$
   
   
   • The probability of not winning C,G is $(1-0.20-0.05)^{15}$
   
   
   • The probability of not winning T,S is $(1-0.30-0.10)^{15}$
   
   
   • The probability of not winning T,D is $(1-0.30-0.35)^{15}$
   
   
   • The probability of not winning T,G is $(1-0.30-0.05)^{15}$
   
   
   • The probability of not winning S,D is $(1-0.10-0.35)^{15}$
   
   
   • The probability of not winning S,G is $(1-0.10-0.05)^{15}$
   
   
   • The probability of not winning D,G is $(1-0.35-0.05)^{15}$
   
   




4. Subtract the following:

   
   
   
   
   
   • The probability of not winning C,T,S is $(1-0.20-0.30-0.10)^{15}$
   
   
   • The probability of not winning C,T,D is $(1-0.20-0.30-0.35)^{15}$
   
   
   • The probability of not winning C,T,G is $(1-0.20-0.30-0.05)^{15}$
   
   
   • The probability of not winning C,S,D is $(1-0.20-0.10-0.35)^{15}$
   
   
   • The probability of not winning C,S,G is $(1-0.20-0.10-0.05)^{15}$
   
   
   • The probability of not winning C,D,G is $(1-0.20-0.35-0.05)^{15}$
   
   
   • The probability of not winning T,S,D is $(1-0.30-0.10-0.35)^{15}$
   
   
   
   • The probability of not winning T,S,G is $(1-0.30-0.10-0.05)^{15}$
   
   
   • The probability of not winning T,D,G is $(1-0.30-0.35-0.05)^{15}$
   
   
   • The probability of not winning S,D,G is $(1-0.10-0.35-0.05)^{15}$
   
   



5. Add the following:

   
   
   
   
   
   • The probability of not winning C,T,S,D is $(1-0.20-0.30-0.10-0.35)^{15}$
   
   
   • The probability of not winning C,T,S,G is $(1-0.20-0.30-0.10-0.05)^{15}$
   
   
   • The probability of not winning C,T,D,G is $(1-0.20-0.30-0.35-0.05)^{15}$
   
   
   
   • The probability of not winning C,S,D,G is $(1-0.20-0.10-0.35-0.05)^{15}$
   
   
   • The probability of not winning T,S,D,G is $(1-0.30-0.10-0.35-0.05)^{15}$
   
   

Please note that the sum of the probabilities is equal to $1$.

If it was smaller, then you would also need to subtract the probability of not winning C,T,S,D,G.

---

Here is a Python script for calculating that:

p = [0.20,0.30,0.10,0.35,0.05]

res = 1

for i in range(0,len(p)):
    res -= (1-p[i])**15
    for j in range(i,len(p)):

        res += (1-p[i]-p[j])**15
        for k in range(j,len(p)):
            res -= (1-p[i]-p[j]-p[k])**15
            for n in range(k,len(p)):
                res += (1-p[i]-p[j]-p[k]-p[n])**15

print res

The result is $0.54837227253$.