9. You want to order 36 Chipotle burritos for your club\'s function. From 11W1 P

Question

9. You want to order 36 Chipotle burritos for your club's function. From 11W1 Problem #8, there are 6 different possible fillings for your burrito. For the purposes of this problem, an order mcans a list of 6 nonnegative integers which count the number of steak, chicken, barbacoa, carnitas, sofritas, and nothing burritos you want. How many orders are there if: (a) you have no further restrictions? (b) for each of the 6 types of fillings, you want at least one burrito of that type? (c) you want exactly 8 chicken burritos? (d) you want at least 3 sofritas burritos and at least 8 nothing burritos, for the vegans and vegetarians? (e) you want at most 4 barbacoa burritos? (f) you want at most 4 barbacoa burritos and at ost carnitas burritos? Those are the most expensive.

Explanation / Answer

There are 6 different types of burritos and you require to buy 36 burritos.

So,

a) There are no restrictions, so each of the 36 burrittos can be chosen in 6 ways possible:

Total number of combinations : 636 ways

b) You need one burrito of each type. The rest 30 can be of any type with no restrictions

So, total number of combinations = ordering 30 burritos with no restrictions

= 630 ways

c)

You need exactly 8 chicken burritos (fixed) and so, the other 28 burritos can be purchased with no restriction

Thus, total number of ways: 528 ways

d)

We need at least 3 sofritas and 8 nothings. So 11 of them are fixed. The rest of them (i.e. 25) can be any bought with combinations having no restrictions.

thus, number of ways : 625 ways

e)

Atmost 4 barbarcoa

= W ( 0 barbarcoa) + W ( 1 barbarcoa) + W ( 2 barbarcoa) + W ( 3 barbarcoa) + W ( 4 barbarcoa)

= 536 + 535 + 534 + 533 + 532 ways

f)

At most 4 barbarcoas and 4 carnitas

Assume, we have 4 of each (i.e to max capacity as allowed, and then we reduce it in each term)

= 428 + 2 * 429 + 3 * 430 + 4 * 431 + 5 * 432 + 4 * 433 + 3 * 434 + 2 * 435 + 436 ways

Hope this helps

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