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Answer to Question #33003 in Algebra for gulmeena

Question #33003
how many 5 digits numbers can be made from the digits 2, 3,4,5,7,9 if no digit is repeated but option (1) digit 2 and 9 are to be next to each other (2) digit 2 and 9 are not to be next to each other?
Expert's answer
1) digit 2 and 9 are to be next to each other :
choosethe place for combinations, where 2 and 9 next to each other - 4 cases.
next toeach other means 29 or 92 - 2 cases.
remainsto choose three digits from 3,4,5,7, as no digit is repeated - for
the third place we can choose one of four digits {3,4,5,7} - 4 cases;
for the fourth plase we can choose fromthree remaining digits (no digit is repeated) - 3 cases; for the fifth ( last) place we can choose from wto
remaining digits.
Use Rule ofproduct, multiply all points : 4 * 2 * 24 = 192.

2)digit 2 and 9 are not to be next to each other
itmeans all 5 digit numbers can be made from the digits 2,3,4,5,7,9 if
no digit is repeated, except numbers from point (1)
to know how much 5 digits numbers can be made from the digits 2, 3,4,5,7,9 ifno digit is repeated:
for the first place we can choose one of six digits {2,3,4,5,7,9} = 6 cases,
for the second place - one of 5 remaining digits - 5 cases, for the third
- 4 cases, for the fourth - 3 cases, for the last - 2 cases.
Use product Rule, multiplyall points and subtract 192 :

6* 5 * 4 * 3 * 2 - 192 = 720 - 192 = 528

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