Question

How many ways can six of the letters of the word ALGORITHM be selected 8. How many ways can the letters of the word ALGORITHM be arranged in a be seated together in the row? and written in a row? row if the letters GOR must remain together (in this order)?

Answer 1

Answer with explanation:

The number of letters in word "ALGORITHM" = 9

The number of combinations to select r things from n things is given by :-

$C(n,r)=\dfrac{n!}{r!(n-r)!}$

Now, the number of combinations to select 6 letters from 9 letters will be :-

$C(9,8)=\dfrac{9!}{6!(9-6)!}=\dfrac{9\times8\times7\times6!}{6!\times3!}=84$

Thus , the number of ways can six of the letters of the word ALGORITHM=84

The number of ways to arrange n things in a row :$n!$

So, the number of ways can the letters of the word ALGORITHM be arranged in a be seated together in the row :-

$9!=362880$

If GOR comes together, then we consider it as one letter,  then the total number of letters will be = 1+6=7

Number of ways to arrange 9 letters if "GOR" comes together :-

$7!=5040$

Thus, the number of ways to arrange 9 letters if "GOR" comes together=5040