Question

Professor Grinch’s telephone number is 6328363. Mickey remembers the collection of digits but not their order, except that he knows the first 6 is before thefirst3.How many arrangements of these digits with this constraint are there?

Answer 1

Answer:

There are 168 arrangements are there as per the conditions

Step-by-step explanation:

Given Number: 6328363

We will use permutation formula which is P = n!/(n-r)!

n represents number of objects taken r at a time

Replace 6s and 3s with Xs:

XXX8XX2

We have $\frac{7!}{5!}$ ways to get the above number,

Now, at first, replace first X with 6 which is always before 3, we get  

6XX8XX2 It means, we only one way of doing this, so this is the only arrangement where we get 6 at first place

Finally we are left with 4 numbers (X's), the arrangements for these 4 numbers are  $\frac{4!}{3!}$

Now the Total Arrangements are

$\frac{7!}{5!}$  * 1 *$\frac{4!}{3!}$ = 42*4 = 168

Answer 2

Answer / Explanation

In this instance as related to the question, we will recall the principle of permutation and combination:

Thus:

If we also recall the telephone number of Professor Grinch that was given, we have:

N = 6328363

Moving forward, we perform the operation below:

Replace 6s and 3s with Xs:

On doing this, we have:

XXX8XX2

Moving forward, we go ahead to perform the operation below:

Replace the first X with 6

On doing the above, we have:

6XX8XX2

On performing the above operation:

We discover that the first 6 will be always before the first 3

giving us XXX8XX2.

Therefore, we can say that we have:

7 ! / 1! 1! 5! number of ways to arrange the number:

6XX8XX2

and 1 way to arrange the number below:

6338632

4 ! /1! 3! ways of arranging the above

Thus, we have:

7 ! / 1! 1! 5! x 1 x 4 ! /1! 3!

= 168