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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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