How many different arrangements are there of the 11 letters in the word REQUIREMENT in which there are exactly three letters between the two Rs? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_s21_qp_53 Year：2021 Question No：6(c)
Answer:
\(
\frac{9 !}{3 !} \times 7
\)
\(
423360
\)
Alternative method for Question 6(c)
\(
{ }^{9} \mathrm{C}_{3} \times 7 !\left(\times \frac{3 !}{3 !}\right)
\)
\(
423360
\)
Alternative method for Question 6(c)
\(
3 \times 7 \times \frac{8 !}{2 !}
\)
423360
Alternative method for Question 6(c)
\( 3 \times 7 \times \frac{8 !}{2 !} \)
\( 7 \times \frac{2}{11} \times \frac{9}{10} \times \frac{8}{9} \times \frac{7}{8} \times \frac{1}{7} \times \) total no. of arrangements
423360
Alternative method for Question 6(c)
No E between the Rs \( -\frac{{ }^{6} C_{3} \times 3 ! \times 7 !}{3 !}=100800 \)
1E between the Rs \( \quad-\frac{{ }^{6} C_{2} \times 3 ! \times 7 !}{2 !}=226800 \)
2Es between the Rs \( \quad-{ }^{6} C_{1} \times 3 ! \times 7 !=90720 \)
3Es between the Rs \( \quad-7 !=5040 \)
\( [\text { Total }=7 ! \times(20+45+18+1)=7 ! \times 84=] 423360 \)
\frac{9 !}{3 !} \times 7
\)
\(
423360
\)
Alternative method for Question 6(c)
\(
{ }^{9} \mathrm{C}_{3} \times 7 !\left(\times \frac{3 !}{3 !}\right)
\)
\(
423360
\)
Alternative method for Question 6(c)
\(
3 \times 7 \times \frac{8 !}{2 !}
\)
423360
Alternative method for Question 6(c)
\( 3 \times 7 \times \frac{8 !}{2 !} \)
\( 7 \times \frac{2}{11} \times \frac{9}{10} \times \frac{8}{9} \times \frac{7}{8} \times \frac{1}{7} \times \) total no. of arrangements
423360
Alternative method for Question 6(c)
No E between the Rs \( -\frac{{ }^{6} C_{3} \times 3 ! \times 7 !}{3 !}=100800 \)
1E between the Rs \( \quad-\frac{{ }^{6} C_{2} \times 3 ! \times 7 !}{2 !}=226800 \)
2Es between the Rs \( \quad-{ }^{6} C_{1} \times 3 ! \times 7 !=90720 \)
3Es between the Rs \( \quad-7 !=5040 \)
\( [\text { Total }=7 ! \times(20+45+18+1)=7 ! \times 84=] 423360 \)
Knowledge points:
5.2.2.1 repetition (e.g. the number of ways of arranging the letters of the word ‘NEEDLESS’)
5.2.2.2 restriction (e.g. the number of ways several people can stand in a line if two particular people must, or must not, stand next to each other). (Questions may include cases such as people sitting in two (or more) rows.) (Questions about objects arranged in a circle will not be included.)
Solution:
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