Yes, she's right.  I didn't count the "G".  I fixed it below:
a)	In how many different ways can the letters of the word KNOWLEDGE be
arranged in such a way that the vowels always come together?
We can choose the 3 vowels to be together (call it a "vowel trio") any of
these 3 ways: EEO, EOE, OEE.

For each of those 3 choices, there are 7 things (six single letters and one
vowel-trio).  The 7 things are

K, N, W, L, D, G, (the vowel trio)

There are 7! ways to arrange those six things:

Answer: 3∙7! = 3∙5040 = 15120 ways.
======================
b) In a group of 6 boys and 4 girls, four children are to be selected. In

how many different ways can they be selected such that at least one boy

should be there?
If we didn't care whether there was at least one boy the answer would be
10 children choose 4 = C(10,4) or 10C4 = 210 ways.

There is just 1 unwanted situation, when the four girls are chosen.  That's
because we can choose all girls in just one way. C(4,4) or 4C4 = 1.  So we
just need to subtract 1 for that 1 unwanted situation:

Answer: 210-1 = 209 ways.

Edwin
  

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
 .


In the word  KNOWLEDGE, we have 6 different consonants and TRIO of vowels.


We consider the trio as one single object (looking from the long distance).


From this point of view, we have 6+1 = 7 objects to permute, which gives 7! = 5040 permutations.



Changing/switching the scope to close view, wee see that the trio of vowels admits 3 arrangements,

giving the total number of all possible arrangements 3*5040 = 15120.



ANSWER.  The total number of all possible arrangements is 15120.




Solved.







 

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