How do you write $66times65times64times27times26$ as a ratio of factorials?

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Answer 1 · 2018-06-24T20:20:42

$(66!)/(63!)*(27!)/(25!)$

We have the following:

$color(purple)(66*65*64)*color(steelblue)(27*26)$

What we have in purple seems to be $66!$ , but it stops at $64$ . How do we account for this?

The key realization is that $66!$ would be

$66*65*64*color(red)(63*62...)$ , but we don't want those extra terms.

If we divide $66!$ by what's in red (essentially $63!$ ), we will be left with $66*65*64$ .

$color(purple)((66!)/(63!))=(66xx65xx64xxcancel(63xx62...))/(cancel(63xx62xx61xx60xx59...))$

We can do the same exercise with $27*26$ . We just want the first two terms of $27!$ , so we divide this $(27-2)!$ , or $25!$ .

$color(steelblue)((27!)/(25!))=(27xx26xxcancel(25xx24...))/cancel(25xx24xx23xx22...)$

We can rewrite $color(purple)(66*65*64)*color(steelblue)(27*26)$ as $color(purple)((66!)/(63!))*color(steelblue)((27!)/(25!))$ .

Hope this helps!

How do you write 66times65times64times27times2666times65times64times27times26 as a ratio of factorials?