Question

Question1: How many of the 5-digit numbers that can be written with the numbers 1, 2, 3, 4 contain both numbers 1 and 2?
question2: 3 married couples shown as a1a2, b1b2, c1c2 want to sit in a row. How many different ways can these married couples sit in a row, without a husband and a wife from the same couple coming together?

Answer 1

1.

ways without using 1 = (no. of options)^5 = 3^5 {no. of options = 3 as we can use 2,3,4}

ways without using 2 = (no. of options)^5 = 3^5 {no. of options = 3 as we can use 1,3,4}

ways without using 1 and 2 = (no. of options)^5 = 2^5 {no. of options = 3 as we can use 3,4}

set A or B = A + B - A and B

therefore

no. of ways to not use 1 or 2 = (ways without using 1) + (ways without using 2) - (ways without using 1 and 2)

= 3^5 + 3^5 - 2^5

= 454

no. of ways to use both 1 and 2 = (total ways to write 5 digit numbers) - (no. of ways to not use 1 or 2 )

= 4^5 - 454

no. of ways to use both 1 and 2 = 570

2.

answer = 240 ways

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