CheesyBakedPasta

Today’s eLetter from the folks at Fine Cooking began “Baked Pasta 259,200 Ways. We did the math.” As you can imagine, I did the Baked Pasta Recipe Maker math, too. I figure it’s 16,128,000 ways, or about 60 times the number Fine Cooking found when they did the math. Here’s the calculation. The recipe maker walked me through the following steps:

  • Choose one or two of four Flavor Bases
  • Chose one of three Sauces
  • Choose two or three of nine Sauce Enhancers
  • Choose one of eight Pastas
  • Choose zero, one, or two of five Vegetables
  • Choose two or three of six Cheeses

Assuming no choice combinations are forbidden (the recipe maker doesn’t appear to prevent you from adding olives and sherry vinegar to sausage and chicken in pink sauce, for example), you find total number of different ways to make a choice at every step by multiplying together the numbers of choices at each step.

It’s easy to count the number of ways to “choose this many of those Things.” If this many is k, and those Things are n in number, the number of ways to choose k of the n things is “n choose k,” sometimes written as C(n,k). These numbers can all be found in Pascal’s triangle. As it’s shown here, C(n,k) is in the row labeled with the n value, under the column labeled with the k value. Here’s how to use the triangle to find the value of C(9,3):Pascal

  • To choose one or two of the four Flavor Bases, there are C(4,1) = 4 ways to choose one plus C(4,2) = 6 ways to choose two, for a total of 10 ways to choose this item.
  • To choose one of the three Sauces, there are C(3,1) = 3 ways.
  • To choose two or three of the nine Sauce Enhancers, there are C(9,2) = 36 ways to choose two plus C(9,3) = 84 ways to choose three, for a total of 120 ways.
  • There are C(8,1) = 8 ways to choose a pasta.
  • There are C(5,0) + C(5,1) + C(5,2) ways to choose up to two vegetables, or 1 + 5 + 10 = 16 ways.
  • There are C(6,2) + C(6,3) to choose the Cheeses, or 15 + 20 = 35 ways

Multiplying these numbers of choices for each step yields 10·3·120·8·16·35 = 16,128,000 ways, about 60 times as many as Fine Cooking found when they did the math. Counting ways isn’t standard recipe math, and I’d like to note that Fine Cooking’s math is generally fine when it comes to ounces, grams, cups, servings, and calories.