Welson's stuffed sandwich

The total number of different sandwiches equals $\binom{2}{1}\cdot\binom{3}{1}\cdot\binom{5}{2}=2\cdot3\cdot10=60$

Assuming that John always eats a different sandwich until he runs out of options, he will eat $7$ sandwiches a week.

Dividing $60$ by $7$ gives $8\frac{4}{7}$ .

Thus, it will be during week $\boxed{9}$ that John eats a sandwich that he has already eaten, since he has enough different sandwiches for $8$ weeks, but not enough for $9$ .

Since John has two types of spices, we have: $2\cdot 3\cdot \binom{5}{2}=60$ Then: $\left \lceil \frac{60}{7} \right \rceil=9$

John has $\dbinom {2}{1} \cdot \dbinom {3}{1} \cdot \dbinom {5}{2} = 2 \cdot 3 \cdot 10 = 60$ possible sandwich combinations, so we will take $\boxed{9}$ weeks to go over $60$ .

3 ingredients

For first, 2 possibilities

For second, 3 possibilities

For Last, 5c2 =10 possibilities

Total ways 2X3X10=60

Total weeks 60/7=8.57

But it has to be an integer, thus we take next integer to 8 i.e. 9

Number of Combinations of sandwiches: $$2\times 3 \times \frac{5\times 4}{2!} = 60$$ Divide by 7 and find the smallest integer greater than the result: $$\left\lceil \frac{60}{7}\right\rceil = 9$$

We have 2 types of bread, 3 kinds of stuffing and 5 types of spices.

From which for making one sandwich, we can choose 1 type of bread in 2 ways , 1 kind of stuffing in 3 ways, 2 types of spices in 5C2 ways i.e., 10 ways.

Therefore, total different breads can be = 2 * 3 * 10 = 60, means all distinct breads can be eaten in 8 weeks and 4 days i.e., John is going to eat the repeated bread in 9th week. (Ans)

Multiply the number of choices of bread by the number of choices of stuffing and the number of choices of spices. The total choices of spices possible is given by ${5}\choose{2}$ = 10.

$2 \times 3 \times 10$ = $\boxed{60}$

let the two breads be b1 and b2, three stuffings be s1, s2, s3 and spices be p1 ,p2,p3,p4,p5one bread can be selected in 2c1 ways stuffing can be selected in 3c1 ways spices can be selected in 5c2 ways. total possible ways are 2c1.3c1.5c2 = 60 hence same sandwhich will repeat after 9 monts(7*9=63)

9