math, help

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How did they solved this problem i got this one wrong on my quizz.

Pizza Varieties A television commercial for Little Caesars
pizza announced that with the purchase of two pizzas, one could receive free any combination of up to five toppings
on each pizza. The commercial shows a young child waiting in line at Little Caesars who calculates that there are
1,048,576 possibilities for the toppings on the two pizzas.*

a. Verify the child’s calculation. Use the fact that Little Caesars has 11 toppings to choose from. Assume that the order of the two pizzas matters; that is, if the first pizza
has combination 1 and the second pizza has combination
2, that is different from combination 2 on the first pizza and combination 1 on the second.

The answer that's in the back of the book: 1, 048,574

b. In a letter to The Mathematics Teacher, Joseph F. Heiser
argued that the two combinations described in part a
should be counted as the same, so the child has actually
overcounted. Give the number of possibilities if the
order of the two pizzas doesn’t matter.

The answer they got was: 524,800


Consider first the number of ways a pizza can be prepared with 0, 1, 2, 3, 4 or 5 toppings selected from leven possible choices. That number is
11!/6!5! + 11!/7!4! + 11!/8!3! + 11!/9!2! + 11 + 1. The "1" represents no toppings at all.
That number is 462 + 330 + 165 + 55 + 11 = 1 = 1024. When two pizzas are made, the number of possible combinations for two (including duplicates) is (1024)62 = 1,048,576

Any differences from that number are probably the result of excluding duplicate pizzas.



That (1024)62 should have been (1024)^2. I didn't hit the shift key hard enough

thank you i got it earlier i wasn't seeing a step...

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