Solutions from a student (dragged) 33

Solutions from a student (dragged) 33 - in ± 5 2 ² ways...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
We can check that this expression is correct by expanding eachs ide . Expand ingthele f t hand side we Fnd that ± 3 n 3 ² = 3 n ! 3!(3 n - 3)! = 3 n (3 n - 1)(3 n - 2) 6 = 9 n 3 2 - 9 n 2 2 + n. While expanding the right hand side we Fnd that 3 ± n 3 ² +3 n 2 ( n - 1) + n 3 =3 n ! 3!( n - 3)! +3 n 3 - 3 n 2 + n 3 = n ( n - 1)( n - 2) 2 +4 n 3 - 3 n 2 = n ( n 2 - 3 n +2) 2 +4 n 3 - 3 n 2 = n 3 2 - 3 n 2 2 + n +4 n 3 - 3 n 2 = 9 n 3 2 - 9 n 2 2 + n, which is the same, showing the equivalence. Problem 10 (counting fve digit numbers with no triple counts) Lets Frst enumerate the number of Fve digit numbers that can b econ s t ru c t edw i thno repeated digits. Since we have nine choices for the Frst digit, eight choices for the second digit, seven choices for the third digit etc. We Fnd the numbero fFved ig itnumbersw ithno repeated digits given by 9 · 8 · 7 ·
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: in ± 5 2 ² ways. To Fll the remaining three digit location can be done in 8 · 7 · 6 ways. This gives in total 9 · ± 5 2 ² · 8 · 7 · 6 = 30240 . Lets now count the number Fve digit numbers with two repeated digits. To compute this we might argue as follows. We can select the Frst digit and its location in 9 · ± 5 2 ² ways. We can select the second repeated digit and its location in 8 · ± 3 2 ² ways. The Fnal digit can be selected in seven ways, giving in total 9 ± 5 2 ² · 8 ± 3 2 ² · 7 = 15120 . We note, however, that this analysis (as it stands) double counts the true number of Fve digits numbers with two repeated digits. This is because in Frst selecting the Frst digit from...
View Full Document

This note was uploaded on 02/25/2011 for the course STAT 418 taught by Professor G.jogeshbabu during the Winter '08 term at Penn State.

Ask a homework question - tutors are online