Meaningless Anime Title

Probability Level 4

Here's my favorite Anime (you may found it in my status):

SHI - GA - TSU - WA - KI - MI - NO - U - SO

Find the number of arrangements of the syllables such that, both WA and NO aren't at the beginning or end or besides each other.

Note: No spoilers in the comment, please.

Clarification: Arrange (not derange ) is to swap positions of the syllables, but you may keep some syllables unchanged.

1 solution

Tran Quoc Dat
Apr 21, 2016

Consider the numbers 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 . We'll put the syllables in replacement with the numbers.

WA and NO can't be on 1 or 9 , so there are $7$ gaps ( 2 - 3 - 4 - 5 - 6 - 7 - 8 ) for these syllables.

If WA is at 2 then NO must be on 4 to 8 ( $5$ ways).

If WA is at 3 then NO must be on 5, 6, 7, 8 ( $4$ ways).

If WA is at 4 then NO must be on 2, 6, 7, 8 ( $4$ ways).

... If WA is at 7 then NO must be on 2, 3, 4, 5 ( $4$ ways).

Lastly, if WA is at 8 then NO must be on 2 to 6 ( $5$ ways).

By Rule of Sum , the number of ways to put WA and NO in the gaps is $5 + 4 \times 5 + 5 = 30$ .

For each way to put the 2 syllables in the gaps, there are $7!$ ways to arrange the other syllables in the other gaps.

Thus, by Rule of Product , there are $30 \times 7! = \boxed{151200}$ ways.

though i got it correct but i recommend that you change

Arrange (not derange) is to swap positions of the syllables, but you may keep some syllables unchanged.

Arrange (not derange) is to swap positions of the syllables, but you may keep some or all syllables unchanged.

aryan goyat - 5 years ago