Answer: (a) \(144\). (b) \(1344\).

(a) Treat the three men as one block, because they must stand next to each other.

Then there are \(4\) objects to arrange: the block of men and the \(3\) women. These can be arranged in

\(4!\)

ways. Within the block, the \(3\) men can be arranged in

\(3!\)

ways. Therefore the total number of arrangements is

\(4!\times3!=24\times6=144.\)

(b) The word has \(4\) vowels and \(9\) consonants. We are choosing \(7\) different letters, and the selection must include at least \(2\) vowels.

Count the possible cases:

exactly \(2\) vowels, exactly \(3\) vowels, or exactly \(4\) vowels.

So the number of selections is

\(\binom42\binom95+\binom43\binom94+\binom44\binom93.\)

Evaluate each term:

\(\binom42\binom95=6\times126=756,\)

\(\binom43\binom94=4\times126=504,\)

and

\(\binom44\binom93=1\times84=84.\)

Hence the total is

\(756+504+84=1344.\)