Permutations And Combinations

Question
CBSEENMA11014174

(a) In how many ways can a girl make a garland with 8 flowers?

(b) In how many ways can a girl make a necklace with 4 red, 3 green and 2 blue beads?

Solution

(a) Number of flowers = 8
     rightwards double arrow                   n = 8
      Number of ways in which the garland can be made
                    space space equals fraction numerator left parenthesis straight n minus 1 right parenthesis factorial over denominator 2 end fraction space equals space fraction numerator 7 factorial over denominator 2 end fraction space equals space 5040 over 2 equals 2520.
(b) Number of beads = (4 red + 3 green + 2 blue) = 9
     Number of ways in which the necklace can be made
                        = fraction numerator begin display style 1 half end style left parenthesis straight n minus 1 right parenthesis factorial over denominator straight p factorial space straight q factorial space straight r factorial end fraction equals 1 half open square brackets fraction numerator 8 factorial over denominator 4 factorial space 3 factorial space 2 factorial end fraction close square brackets
                        =  1 half cross times fraction numerator 8 cross times 7 cross times 6 cross times 5 cross times 4 cross times 3 cross times 2 cross times 1 over denominator left parenthesis 4 cross times 3 cross times 2 cross times 1 right parenthesis space left parenthesis 3 cross times 2 cross times 1 right parenthesis space left parenthesis 2 cross times 1 right parenthesis end fraction
                        = 70.

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