Rosen 7e Exercise-8.5 Question-15 page no-558 Inclusion-Exclusion

Question

How many permutations of the 10 digits either begin with the 3 digits 987, contain the digits 45 in the fifth and sixth positions, or end with the 3 digits 123?

Comments

5158 ?

Answer 1

numbers starting with 978 => 987_ _ _ _ _ _ _ => 7!
numbers having 45 in 5th and 6th position => _ _ _ _ 45_ _ _ _ => 8!
numbers ending with 123=7!

numbers starting with 978 and having 45 in 5th and 6th position=98745_ _ _ _ _ =>5!
numbers starting with 978 and ending with 123 => 978 _ _ _ _ 123 =>4!
numbers having 45 in 5th and 6th position and ending with 123 => _ _ _ 45 _ _ 123 =>5!

numbers starting with 978, having 45 in 5th and 6th position and ending with 123 => 98745_ _ 123 =>2!

7!+8!+7!-5!-4!-5!+2!

=50,138

Answer 2

$($begin with the 3 digits 987, contain the digits 45 in the fifth and sixth positions$)$ $or$ $($end with the 3 digits 123 $)$

let the $10$ digits be represented as _ _ _ _ _ _ _ _ _ _

987_ 45_ _ _ _  = $5!$

_ _ _ _ _ _ _ 123 = $7!$

987_45_ 123   = $2!$

$N(A\cup B)$

= $N(A)+N(B)-N(A\cap C)$

= $5! +7! -2!$

=$120 +5040 - 2$

=$5158$

Comments

i think u interpreted it wrongly

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So what would have been written in the question for the way which i have solved ? and how did you came to know that there are three cases not two ? @aditi19

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nothing could be written. The question is itself slightly ambiguous