# Keeping track of the factors

By Bobby Neal Winters

You may recall my having told you about a person named Wilma B. Even in a piece entitled “Uncountable Candy.” Wilma is the president of the United Methodist Women in the First United Methodist Church in Kimberly, Kansas.

By the way, don’t look for Kimberly, Kansas on any map because it has so far avoided the attention of any cartographer. It was christened Kimberly after Kimberley, North Cape, South Africa, a place where diamonds were discovered. It’s in Kansas, there were never any diamonds discovered there, and the folks who named it left out the last “e” because it didn’t look to them like it was actually needed. They are a parsimonious lot in Kimberly.

In my mind, the name fits, however, because the people there are such jewels. Wilma is one of those jewels and personifies the spirit of the place better than anyone else I know.

The parsimony of Kimberly is a result of its location in what in many ways is still a frontier. As one looks west from Kimberly, except for the occasional town and small city, one sees a gradual thinning of population, a great emptiness. One gets the feeling there is no one around to help, and one learns not to count on help. This argues against waste. It argues against change.

Wilma is a product of this environment and one of its best exemplars. One of her characteristics is that she finds a way to do something and then falls into the trap that it is the only way or the best way. Such was the case during her church’s Vacation Bible School a few years ago.

It had been her church’s custom to give the children different gifts on each day of the week: on Monday they would receive crayons; on Tuesday coloring books; on Wednesday scissors; on Thursday glue; and on Friday they would receive glitter.

Looking at the list, there is a certain logic in it. I can envision a reality in which there would’ve been reasons associated with giving the gifts in this precise order. Indeed, I’ve been shown the curriculum the church used to use and there were needs for particular items on particular days. Due to the realities of the publishing industry, they’ve been forced to switch to a particular curriculum in which the children having their own scissors earlier in the week would be handy.

As with so many things that come my way by virtue of having a wide circle of acquaintances, I do not know exactly how everything occurred, and, what is more, I don’t want to know. I have been told there was an energetic discussion in which one of the Vacation Bible School teachers informed Wilma that there were a lot of different ways to do this. Wilma disagreed with that—energetically—and words were exchanged.

Let me emphasize that I knew nothing, and I mean nothing, of this when Wilma approached me. I am a friend of Wilma’s, and being Wilma’s friend has a certain amount of overhead associated with it.

We had met at an event—I don’t remember which one—in which donuts and coffee were being served. Before I had time to notice, she had laid out the situation involving giving out the gifts in a very neutral way and then proceeded cagily.

“How many ways are there of doing that?” she asked. “Would it take too long for you to figure that out?”

I had no idea I was being asked a controversial question.

“It wouldn’t take long at all,” I said. “There are 120 ways to do it.”

She seemed somewhat surprised with the rapidity of my answer.

“How did you figure that out so quickly?” she asked. “I am not sure I trust an answer that you can get so quickly.”

“Oh,” I said, “this is something that I teach almost every semester. When you make an ordered list of five items there are five factorial ways of putting those items in order. Five factorial is equal to 120.”

She was beginning to look a little grim.

“I am not sure what you are saying,” she said. “Five factorial?”

“That’s right,” I said. “Five factorial is five times four times three times two times one. That is equal to 120.”

“Okay,” she said, “I understand that five times four times three times two times one is equal to 120, but why is that the number of ways of putting five things in order.”

I’d just finished my donut and had a napkin in my hand that was no longer serving a purpose, so it was at that point I pulled my pen from my pocket and drew a diagram on the aforementioned napkin that looked something like this.

“For the sake of time,” I began, as I showed the napkin to her, “I’ve decided to consider the case only for the first three days. I am sure that this will be enough for you to see the pattern.”

We mathematicians are optimistic creatures and have confidence that in following a pattern three times is enough for anyone to get it. As with most optimists, we are constantly disappointed. Nevertheless, she nodded her head that she was willing to buy into the proposition.

“On the first day, there are three choices: crayons, books, and scissors.”

She nodded, so I continued.

“Suppose that we choose crayons as our first choice. This leaves only books and scissors to choose from.”

Again she nodded.

“Then suppose on the second day we choose books. This leaves scissors as the only thing left to choose on the third day. That path along the top in my graph represents that particular series of choices.”

She nodded again.

“Noticed that this graph I’ve drawn branches.”

Again there was a nod.

“This is because there are choices to be made. On the first day there are three alternatives, on the second day there are two, and on the third day there is only one choice. Notice that we each sequence of choices corresponds to one of the paths in the graph. The number of paths can be obtained by multiplying three by two by one. If we go the whole week, it is the same principle.”

She wasn’t looking too happy, but this isn’t too unusual in people I explain math to, so I am used to it, but she was still listening to me. That, I am not used to.

“You call this ‘factorial’?” she asked.

“Yes,” I said. “You may have seen it written out as an exclamation mark.” I turned over the napkin and wrote it out.

3!=6

5!=120

7!=5040

“See, they get quite big,” I said. “That takes a lot of people by surprise.”

“Yes, it does,” she said.

I didn’t learn until later the story behind it all.

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