Permutations and Combinations Test - 7

The word " LAUGHTER" has  consist of  8 letters which are all distinct

Hence the required number of arrangements=⁸P₈ = 8! = 40320

We have 36000 = 2⁵3²5³

To find the even factors  we should choose atleast one 2's from the slot ,which will ensure that whatever the remaining selection the multiplication of those will always results an even number.

The number of ways to select atleast one 2's from a lot of 5 identical 2's will be 5 ways(select 1,select 2,select 3,select 4,select 5)

And we can select any number of 5's from a lot of three identical 5's in 4 ways(select 0,select 1,select 2) and similarly we can select any number of 3's from a lot of two identical 3's in 3 ways

Therefore total number of selection of even factors = 5 × 4 × 3 = 60 different ways

First we will find the number of four digit numbers (i.e, numbers from 1000 to 9999) which can be formed using the digits 0,1,2,3,4,5,6,7,8 and 9 with repitition allowed.

Now we have the first place can be filled by any of the 9 digits other than 0 and since repetition is allowed second ,third and fourth can be filled by any of the ten digits.

Hence total number of four digit numbers = 9 × 10 × 10 × 10 = 90009

Now we will consider the case that the number does not have the digit '7'

Now the first place can be filled by any of the 8 digits other than 0 and  7 since repetition is allowed second ,third and fourth can be filled by any of the 9 digits other than 7.

Hence total number of ways we can form a four digit number with out 7 = 8 × 9 × 9 × 9 = 58328

Hence total number of ways in which we can form a four digit number having atleast one digit as 7=9000−5832=3168

Since we have to form a four digit number using the digits 0,2,3,5,8,9 , we can fill the first place by any of the five digits given other than zero.

Now since the digits can be repeated the second ,third and , fourth places can be filled by any of the six given digits

Hence the four places can be filled together in 5 × 6 × 6 × 6 = 1080 different ways

Therefore the number of four digit numbers which can be formed using the given digits =1080

Since all the plus signs are identical ,  we have number of ways in which  5 plus signs  can be arranged=1.

Now we will have 6 empty slots between these 5 identical + signs

Hence number of  possible places of - sign =6

Therefore number of ways in which the 5  minus sign  can take any of the possible 6 places=C(6,5)