Ihave 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 different pairs of strings can i make?

Solution:

There are four general types we can make: (6,12),(7,11),(8,10),(9,9).

First type: (6,12), there are 18 possible ways to choose those 6, which are going to be cut from the necklace: choose a direction to count the beads, and choose a starting position (between 2 beads). There are exactly 18 starting positions, since there are 18 spaces between the beads. One cut is equivalent for a pair we can make.

Second type: (7,11), with the same reasoning, there are 18 possible ways to cut the necklace.

Third type: (8,10), with the same reasoning, there are 18 possible ways to cut the necklace.

Fourth type: (9,9), The same reasoning cannot be applied again, since half of the cuts would be exactly the same as the other half. So there are 9 possible cuts, exactly one or each axis of symmetry for the necklace.

The solution will be the sum these values: 18+18+18+9=63

first, we define the variables:

x: number of years after 1950

f (x): amount of vinyl sold.

then, with the variables defined, we have:

68594 vinyl records were sold in 1958 > f (8) = 68594

91299 vinyl records were sold in 1961 > f (11) = 91299

38720 vinyl records were sold in 1952 > f (2) = 38720

161743 vinyl records were sold in 1967 > f (17) = 161743

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step-by-step explanation:

x-3y=4

step-by-step e