Answer
$6,302,555,018,760$
Work Step by Step
There are $365$ choices for the first birthday, $364$ choices for the second birthday, $363$ choices for the third birthday, $362$ choices for the fourth birthday, $361$ choices for the fifth birthday. Therefore,
$365 \times 364 \times 363 \times 362 \times 361 = 6, 302, 555, 018, 760$