Question 1164753
<pre>
Yes, she's right.  I didn't count the "G".  I fixed it below:
<pre>
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?
<pre>
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.
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</pre>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?<pre>
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</pre>