[A combinatorial approach to Question 11] In this question we consider binary words consisting only of the letters A and B. a Consider a binary word consisting of a + b letters, with A occuring a times and B occuring b times. Show that there are a+*Ca = a+*C» permutations of such a word. b How many possible permutations are there of a binary word with 2n letters, if A and B both occur n times? © Bill Pender et al. 2019 Photocopying is restricted under law and this material must not be transferred to another party Cambridge University Pr Extension 1 Year 11 ISBN 978-1-108-46907-4 aths Stage 6 15D Identities in Pascal's triangle c A word with 2n letters may be split down the middle into two words of n letters. Consider the example where two As fall in the first n-letter word. i How many arrangements are there of the first n-letter binary word with two As? ii How many arrangements are there of the second n-letter binary word with n - 2 of the As and 2 Bs? iii How many arrangements are there of a ten-letter binary word with two As in the first half and two Bs in the second half? Hence prove that d + ("C2n)? ("Co)² + ("C1)² + ("C2)² + ("C3)² + = 2"Cn- + . ..
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d
![[A combinatorial approach to Question 11]
In this question we consider binary words consisting only of the letters A and B.
a Consider a binary word consisting of a + b letters, with A occuring a times and B occuring b times.
Show that there are a+*Ca = a+*C» permutations of such a word.
b How many possible permutations are there of a binary word with 2n letters, if A and B both occur n
times?
© Bill Pender et al. 2019
Photocopying is restricted under law and this material must not be transferred to another party
Cambridge University Pr
Extension 1 Year 11
ISBN 978-1-108-46907-4
aths Stage 6
15D Identities in Pascal's triangle
c A word with 2n letters may be split down the middle into two words of n letters. Consider the
example where two As fall in the first n-letter word.
i How many arrangements are there of the first n-letter binary word with two As?
ii How many arrangements are there of the second n-letter binary word with n - 2 of the As and
2 Bs?
iii How many arrangements are there of a ten-letter binary word with two As in the first half and two
Bs in the second half?
Hence prove that
d
+ ("C2n)?
("Co)² + ("C1)² + ("C2)² + ("C3)² +
= 2"Cn-
+
.
..](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F211fd0d2-45cd-4603-b99f-222084379316%2F2db7ae6b-dc21-4a7b-b578-7dea58aae101%2Fyki181m.png&w=3840&q=75)
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