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Question-181251




Question Number 181251 by cortano1 last updated on 23/Nov/22
Commented by cortano1 last updated on 23/Nov/22
 A photograph is to be taken of seven girls    standing in two rows, with one row  in front of the other. The tallest girl   must be in the midlle of the row of  three and the shortest girl must be  in row at the front. In how many  ways can the girls be arranged ?
$$\:\mathrm{A}\:\mathrm{photograph}\:\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{taken}\:\mathrm{of}\:\mathrm{seven}\:\mathrm{girls}\: \\ $$$$\:\mathrm{standing}\:\mathrm{in}\:\mathrm{two}\:\mathrm{rows},\:\mathrm{with}\:\mathrm{one}\:\mathrm{row} \\ $$$$\mathrm{in}\:\mathrm{front}\:\mathrm{of}\:\mathrm{the}\:\mathrm{other}.\:\mathrm{The}\:\mathrm{tallest}\:\mathrm{girl}\: \\ $$$$\mathrm{must}\:\mathrm{be}\:\mathrm{in}\:\mathrm{the}\:\mathrm{midlle}\:\mathrm{of}\:\mathrm{the}\:\mathrm{row}\:\mathrm{of} \\ $$$$\mathrm{three}\:\mathrm{and}\:\mathrm{the}\:\mathrm{shortest}\:\mathrm{girl}\:\mathrm{must}\:\mathrm{be} \\ $$$$\mathrm{in}\:\mathrm{row}\:\mathrm{at}\:\mathrm{the}\:\mathrm{front}.\:\mathrm{In}\:\mathrm{how}\:\mathrm{many} \\ $$$$\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{girls}\:\mathrm{be}\:\mathrm{arranged}\:? \\ $$
Answered by mr W last updated on 23/Nov/22
case 1:  XXXX  STX  ⇒2×5!  case 2:  XTX  SXXX  ⇒4×5!  totally: 6×5!=720 ways
$${case}\:\mathrm{1}: \\ $$$${XXXX} \\ $$$${STX} \\ $$$$\Rightarrow\mathrm{2}×\mathrm{5}! \\ $$$${case}\:\mathrm{2}: \\ $$$${XTX} \\ $$$${SXXX} \\ $$$$\Rightarrow\mathrm{4}×\mathrm{5}! \\ $$$${totally}:\:\mathrm{6}×\mathrm{5}!=\mathrm{720}\:{ways} \\ $$

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