Final answer:
The probability that birds of a feather flock together on the telephone wire is 0.009.
Explanation:
In order to find the probability that birds of a feather flock together, we first need to determine the total number of ways in which the birds can be arranged on the telephone wire. Since there are 14 birds in total and they are sitting in a random order, the number of ways they can be arranged is given by 14! (read as 14 factorial). This can also be written as 14x13x12x11x10x9x8x7x6x5x4x3x2x1.
Next, we need to determine the number of ways in which all the three crows can be arranged among themselves on the wire. This can be done in 3! (read as 3 factorial) ways. Similarly, the 5 blue jays can be arranged in 5! ways and the 6 starlings can be arranged in 6! ways.
Therefore, the total number of ways in which birds of the same type can be arranged together is given by 3! x 5! x 6!. This can also be written as (3x2x1) x (5x4x3x2x1) x (6x5x4x3x2x1).
Hence, the probability that birds of a feather flock together is given by:
= [3! x 5! x 6!] / 14!
= (3x2x1) x (5x4x3x2x1) x (6x5x4x3x2x1) / (14x13x12x11x10x9x8x7x6x5x4x3x2x1)
= [6 x (120) x (720)] / (14x13x12x11x10x9x8x7x6x5x4x3x2x1)
= 518400 / 87178291200
= 0.000005943
= 5.943 x 10^-6 (in scientific notation)
= 0.009 (rounded to three decimal places)
Therefore, the probability that birds of a feather flock together on the telephone wire is 0.009.