A box has two compartments separated by a removable gate.  A number of neon atoms are put into the first compartment of the box.  a second number of helium atoms are put into the second compartment.  The gate between the two compartments is lifted and the atoms are allowed to mix.  Then the gate is replaced and the atoms in each compartment are counted.  It is found that the first compartment contains only neon atoms and that the second compartment contains only helium atoms.  The probability of this occurring depends on the number of neon atoms (N) and helium atoms  (H) as well as the size of the two compartments (SN and SH).

The probability is given by the following formula:

Note:  The denominator is the binomial coefficient that we studied in section 10.7
DO NOT TRY TO CANCEL THE FACTORIALS WHICH RESULT FROM THE BINOMIAL COEFFICIENT--THE NUMBERS ARE TOO LARGE

Follow the steps below to find the probability.  I want the answer to be expressed in scientific notation.

1.  To find the numbers of neon and helium atoms with which you are working:

Let the value of N be what you get when you look up the first letter of your last name in the table below.
Let the value of H be what you get when you look up the second letter of your last name in the table below.
 
 

A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X	Y	Z
100	200	300	400	500	600	700	800	900	1000	1100	1200	1300	1400	1500	1600	1700	1800	1900	2000	2100	2200	2300	2400	2500	2600

2.  To find the sizes of the compartments with which you are working:

Let the value of SN be what you get when you look up the first letter of your last name in the table below.
Let the value of SH be what you get when you look up the second letter of your last name in the table below.
 
 

A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X	Y	Z
10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35

N= ________    H = __________   SN = __________  SH = ___________
 
 

3.  Evaluate the formula above with the values of N, H, SN, SH that you found above.

4.  Express the binomial coefficient in term of factorials.

5.  Simplify by inverting and multiplying.

6.  Take the natural logarithm of both sides of the equations above.

7.  Use the laws of logarithms to expand.

8.  Rewrite using Stirling's Approximation where appropriate:   ln(x!) = x ln(x) - x

9.  Simplify again.

10.  Find the values of the logarithms with your calculator.  Don't round off at any stage of the problem.  Keep all the decimals that your calculator will hold in the display.

11.  Get a final value for the computation (which equals  ln P).

12.  From  ln P , use the change of base theorem to find  log P.

13.  Find  P in exponential form.

14.  Use the laws of exponents to find the power part of the scientific notation.

15.  Use your calculator to find the coefficient part of the scientific notation.

16.  Make sure that your answer is in scientific notation.

Examples of answers which are all equivalent:
6.023 (10)²³   is in scientific notation--decimal after the first digit).
60.23 (10)²²   is not in scientific notation.
.6023 (10)²⁴   is not in scientific notation.
10^23.779813  is not in scientific notation.
10²³10^.779813  is not in scientific notation.