## THE GLASS BALLS.

(

Combination and Group Problems)

A number of clever marksmen were staying at a country house, and the

host, to provide a little amusement, suspended strings of glass balls,

as shown in the illustration, to be fired at. After they had all put

their skill to a sufficient test, somebody asked the following question:

"What is the total number of different ways in which these sixteen balls

may be broken, if we must always break the lowest ball that remains on

any string?" Thus, one way would be to break all the four balls on each

string in succession, taking the strings from left to right. Another

would be to break all the fourth balls on the four strings first, then

break the three remaining on the first string, then take the balls on

the three other strings alternately from right to left, and so on. There

is such a vast number of different ways (since every little variation of

order makes a different way) that one is apt to be at first impressed by

the great difficulty of the problem. Yet it is really quite simple when

once you have hit on the proper method of attacking it. How many

different ways are there?

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