I have a circular necklace with #18# beads on it. All the beads are different. Making two cuts with a pair of scissors, I can divide the necklace into two strings of beads. If I want each string to have at least #6# beads, how many differe?

...how many different pairs of strings can I make?

1 Answer
Somebody N.
Jun 17, 2018

#color(blue)(236912)#

Explanation:

If we have 18 beads altogether, and after cutting no string has less than 6 beads, then we can choose from 6 at a time to 12 at a time:

We can only choose up to 12 at a time, since the remaining string must have at least 6 beads.

#:.#

#18-6=12#

So we want the number of combinations of:

#color(white)(0)^18C_6+color(white)(0)^18C_7+color(white)(0)^18C_8+color(white)(0)^18C_9+color(white)(0)^18C_10+color(white)(0)^18C_11+color(white)(0)^18C_12#

#18564+31824+43758+48620+43758+31824+18564=236912#

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