Permutations And Combinations

Question
CBSEENMA11014203

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has:
(i) no girl (ii) at least one boy and one girl (iii) at least 3 girls.

Solution

Total number of boys and girls in the group = 4 girls + 7 boys = 11
Number of boys and girls in the team = 5
(i)      The team consists of no girl,
rightwards double arrow    The team consists of 0 girl + 5 boys
∴        Number of selections = straight C presuperscript 4 subscript 0 cross times straight C presuperscript 7 subscript 5 space equals space 1 space cross times space fraction numerator 7 cross times 6 cross times 5 factorial over denominator 5 factorial space 2 factorial end fraction equals 21
Hence, the number of teams formed = 21
(ii) The team consists of at least 1 boy and 1 girl.
Options are:
      The team consists of 1 girl + 4 boys
                      Number of selections = space space straight C presuperscript 4 subscript 1 cross times straight C presuperscript 7 subscript 4 space equals space 4 cross times fraction numerator 7 factorial over denominator 3 factorial space 4 factorial end fraction equals 4 cross times fraction numerator 7 cross times 6 cross times 5 over denominator 3 factorial end fraction
                                                  = 4 x 7 x 5 = 140.
                                            Or
The team consists of 2 girls + 3 boys
rightwards double arrow                 Number of selections = straight C presuperscript 4 subscript 2 cross times straight C presuperscript 7 subscript 3 space equals space fraction numerator 4 factorial over denominator 2 factorial space 2 factorial end fraction cross times fraction numerator 7 factorial over denominator 4 factorial space 3 factorial end fraction
                                                 = fraction numerator 4 cross times 3 over denominator 2 cross times 1 end fraction cross times fraction numerator 7 cross times 6 cross times 5 over denominator 3 cross times 2 cross times 1 end fraction equals 6 cross times 35 equals 210.
                                          Or
The team consists of 3 girls + 2 boys
rightwards double arrow                  Number of selections = straight C presuperscript 4 subscript 3 cross times straight C presuperscript 7 subscript 2 space equals space fraction numerator 4 factorial space over denominator 3 factorial space space 1 factorial end fraction cross times fraction numerator 7 factorial over denominator 5 factorial space space 2 factorial end fraction
                                                  = 4 cross times fraction numerator 7 cross times 6 over denominator 2 end fraction equals 84.
                                          Or
The team consists of 4 girls + 1 boy.
rightwards double arrow                Number of selections = straight C presuperscript 4 subscript 4 cross times straight C presuperscript 7 subscript 1 space equals space 1 cross times 7 equals 7
Hence, the total number of teams that can be formed = 140 + 210 + 84 + 7 = 441
(iii) The team consists of at least 3 girls
Options are:
 The team consists of 3 girls + 2 boys.
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                                        Or
   The team consists of 4 girls + 1 boy.
rightwards double arrow          Number of selections = straight C presuperscript 4 subscript 4 cross times straight C presuperscript 7 subscript 1 space equals space 1 cross times 7 space equals space 7
Hence, the total number of teams that can be formed = 84 + 7 = 91
   

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