in how many ways can we arrange the letters of the word "boy"taking
two letters at a time

Hello and thank you for your question.

Here's the 'brute force' answer: six
bo ob
by yb
oy yo

Technically, this is an example of a "Permutation of less than all,"
that is,  number of permutations of 3 different things taken 2 at a
time.

This is usually symbolized as  3P2, or 3!/(2-1)! or (3*2*1)/1 = 6

Here is how Dr. Math explains it (Dr. Math uses CAT as the test word)

"For our CAT example, we have:

          3!     3 x 2 x 1
 3_P_2 = ---- = ----------- = 6
          1!         1

We can use any one of the three letters in CAT as the first member of
a permutation. There are three choices for the first letter: C, A, or
T. After we've chosen one of these, only two choices remain for the
second letter. To find the number of permutations we multiply: 3 x 2 =
6.

Note: What's a factorial? A factorial is written using an exclamation
point - for example, 10 factorial is written 10! - and means multiply
10 times 9 times 8 times 7... all the way down to 1."

http://www.mathforum.com/dr.math/faq/faq.comb.perm.html

Search terms used:
 permutation word letters order

That's it!

Google Answers Researcher
Richard.ga

6

choose 3 elements by 3C2 and then arrange them by 2!

3C2*2!=6