Bunuel wrote:

Sid intended to type a seven-digit number, but the two 3's he meant to type did not appear. What appeared instead was the five-digit number 52115. How many different seven-digit numbers could Sid have meant to type?

A. 10

B. 16

C. 21

D. 24

E. 27

Kudos for a correct solution.

Should be 21.

there are two possibilities for placing 2 3s .

case 1: two 3s were missed consecutively. i.e. he typed 33 and it came blank on screen.

-5-2-1-1-5- in this arrangement we can fit 33 in 6 ways . (Six dashes, each dash represent one possible place for placing 33)

case 2: two 3s are not together, i.e. they have one or more digits between them .

-5-2-1-1-5- , in this arrangement

if we place first 3 at first dash i.e. 35-2-1-1-5- then the other 3 can fit into 5 places.

if we place first 3 at second dash i.e. -532-1-1-5- then the other 3 can fit into 4 places.

if we place first 3 at third dash i.e. -5-231-1-5- then the other 3 can fit into 3 places.

if we place first 3 at fourth dash i.e. -5-2-131-5- then the other 3 can fit into 2 places.

if we place first 3 at Fifth dash i.e. -5-2-1-135- then the other 3 can fit into 1 place.

so total 15 ways.

case 2 + case 1 = 6+ 15 = 21 ways

Answer C