Three Elementary Counting Problems

Part I

Three counting problems
In how many ways can a photographer pose seven students in shoulder-to-shoulder alignment?

Two of the seven students adamantly refuse to be photographed side-by-side. In how many ways can a photographer pose the seven students respecting the students’ wishes?

The partners of a second couple announced they do not wish to be photographed side by side. If a photographer honors these two couples wishes, how many ways can the seven students be posed?

Prologue
This tutorial discusses the solution of three counting problems. The techniques discussed exposes secrets that are useful to solve many counting problems.

Are there formulas to remember? Yes! And they should be memorized!

But unlike your past mathematics experiences, substituting numbers into a formula is not the best method to solve counting problems.

A promising approach to solving counting problems is to break the given problem into simpler problems whose solutions lead to the solution of the more difficult given problem. Two fundamental counting principles are discussed. They are helpful in breaking a given problem into simpler problems for solution to the givnen problem.

The first of the two fundamental principles is the addition rule.

The addition rule
If there are m tags in an urn and n tags in a second urn then the total number of ways of selecting a tag from the first urn or the second urn is (m+n).

The second fundamental counting principle is a multiplicative rule, hence it is called the product rule.

Example: A student must select one book from the library’s collection of complexity theory or one book from l collection of Mozart’s biographies. The library has 5 books in its complexity collection and it has 20 biographies of Mozart. The addition rule asserts there are (5+20) ways to select a book from one of these two library collections.

To apply the addition rule,  for each complexity book, put its title on a tag and place the tag in an urn. Into a second urn, place a tag containing the name of each biography. The addition rule asserts that the total number of ways to select a book from the first collection or the second collection is (5+20).

The addition rule can be used in a different way. Consider this example.

Mary registered her upcoming wedding with Stacy Department Store. Stacy’s supplies 147 different plate designs of which 94 of the designs have non-circular plates. Mary likes only circular plates. How many choices does Mary have available through a Stacy’s registration?

In this example a tag describes a plate featured at Stacy’s. There are 94 tags identifying a non-circular plate. Denote by “n”, the number of Stacy’s circular plates. By the addition rule, 147=94+n. Hence, there are (147-94) circular plates available to Mary through a Stacy’s registration. The two urns are those containing the tags of circular plate designs and containing the tags of non-circular plate designs.

The product rule
If an urn contains m tags and a second urn contains n tags then there are m*n ways to form a list of two tags, where the first tag in the list comes from the first urn and the second tag in the list comes from the second urn.

A list is a presentation of objects along a line. Lists are ordered. They have a first object, a second object, etc.

Example:
Urn 1 contains three tags, each tag bearing one of the digit 1, 2, or 3.
Urn 2 contains two tags. One of the tags is marked “a” and the other tag is marked “b” .

The product rule asserts that there are 3*2=6 two-tagged lists, where the first tag is from Urn 1 and the second tag is from Urn 2.
The six lists are: 1a, 2a, 3a, 1b, 2b, 3b, 1c, 2c, 3c .

A Reminder: The product rule prescribes order within the lists. The first tag of the list is from the Urn 1 and the second tag of the list is from Urn 2. This ordering must be honored.

The above reminder asserts that 1c and c1 are different in that two-tagged list. In fact, c1 cannot occur from the order imposed in the construction of the list.

The two counting rules can be extended to more than two conditions (urns).

Example
Given three urns U1 having three tags, U2 having 4 tags, and U3 having 6 tags. No two urns share a tag. The number of ways of selecting a tag from U1 or U2 or U3 is (3+4+6)=13.

The Extended Addition Rule
Given k urns: U1, U2, …, Uk. U1 contains n1 tags, U2 contains n2 tags, …, and Uk contains nk tags such that no two urns share a tag then the total number of ways of selecting a tag from U1 or U2, …, or Uk is (n1+n2+…+nk).

Here n1 is used to represent the number of tags in the first urn. n2, n3, etc. is similarly defined.

The Extended Product Rule
Given k urns: U1, U2, …, Uk. U1 contains n1 tags, U2 contains n2 tags, …, and Uk contains nk tags then the total number of lists of length k with the first tag from U1, the second tag from U2, …, and the kth tag from Uk is the product n1*n2*…*nk.

Suggestion: Memorize the counting rules. A helpful strategy that aids memorizing the counting rules is to recite them to yourself each time you use them. They are the means through which a problem will be broken into smaller problems that leads to the solution to more complicated problems.

Problem 1: In how many ways can a photographer pose seven students in shoulder-to-shoulder alignment?

For the solution, follow this link.

A Black History Commemoration

(The polite terms of the period are used in this article.)

Fannie Lou Hamer, Martin Luther King, Jr., Thurgood Marshall, Madame C. J. Walker, W. E. B. DuBois, Booker T. Washington, Benjamin Bannecker, George Liele are all noteworthy selections that should be cited during Black History Month. Their accomplishments were stellar in the History of Americans of African descent. Their stories are inspiring. Their commitment and fortitude are noteworthy models to be used in future struggles against injustice.

Notwithstanding their noble accomplishments, this writing focuses on a sisterhood that spanned more than 90 years of American history. Its members are linked by their common choice to relinquish what little joy life may have offered, in order to enhance the lives of family members and friends. They were the washwomen of the Negro race in America. The times were the early days of the Negro’s struggle for dignity and freedom in the United States of America.

Carter G. Woodson first called attention to them in The Negro Washwoman, a Vanishing Figure. Venerable saints – he called them. Saints they were; seldom venerated; more often forgotten. This note retells their story and calls for an annual remembrance of their life-struggles – especially their progenies and disciples of compassion.

Their Story: Pre-Emancipation
Their story began in the South during the ante bellum era. They were in service to an often-truculent plantation mistress. Their jobs were to keep their mistresses’ lives free of drudgery. At dawn, they left their hut to prepare their mistress’ meals, to toil with their mistress’ children, to wash and iron their mistress’ household clothes and to clean their mistress’ houses, while their home went unattended until they returned at dusk.

Upon their return, they found their children craving  mother’s love and attention; and their tired husbands resting from the drudgeries of fieldwork and the indignities of their life-status. Without regard for self, these ladies launched into the second half of their day’s toil. But this time, they were cooking and caring for their loved ones, thereby, stoking the flame that fueled their raison d’etre.

The final task of her day our saint devoted to her home enterprise; washing clothes for pay. From whence springs the term, washwoman. With this income, she purchased presents and clothes for her family, and sometimes, household items that turned her hut into a home. Amidst servitude, to become a washwoman was to surrender self for the good of others. It was a life commitment to elevate her family from mere existing, to a life with occasional joy. Our saint ended her day with less than a six-hours rest, ever knowing, the next weekday promised a rerun of the same.

The ante bellum system was taxing to the slaves, but to the female slave it was most agonizing. Slave marriages were prohibited by law, so masters encouraged their female slaves “to take up” with a man. Absent love, she refused, but the master’s will prevailed. Some of these “unions” were lasting. But their purposes were to enlarge the master’s slave inventory for auction and maintenance of his slave force for service. This made birthing a child into slavery tortuous, but it made separating from that child to support such an economic system an agonizing abomination.

In the North the freed colored woman fared better. She did not have to contend with the problems of slavery. But her economic plight was no different. If her husband had no trade, he worked menial jobs for low pay. Many able wives without artistic talent opted to supplement their husband’s low wages by “taking in wash” for pay.

Their Story: Post-Emancipation
Emancipation changed everything. Black men withdrew their wives and daughters from the fields to work at home. Many former slaves migrated to the North to seek a better life. The defeated South was in economic disarray. Returning soldiers were granted the skilled jobs, leaving the menial jobs with less-than subsistence pay to the emancipated. This prevalent scenario forced emancipated females to supplement their family income. Without education or artistic skills, washing clothes in her home was the choice of many: the washwoman.

To many of the emancipated, the North promised more than it delivered. Recently freed males were highly skilled from their trade experience on plantations. Unions closed their rolls to the emancipated to protect their members from these highly skilled competitors, leaving for them menial jobs with very low pay. Freedom in the North had changed the slaves’ social status, but it failed to impact their economic viability.

In the North, the emancipated female did not have to contend with the command  “take up with a man” nor did she have to see her child on an auction block. But her family experienced economic hardship equal to that of her Southern counterpart.  For her life’s trials merely changed its face from the demanding mistress or the dictating master, to that of economic reality in a cruel world: a nonworking husband or one with low pay.

After emancipation, the washwomen continued their sacrifices over 90 years. Then demand for their services gave way to modern wash machinery, modern laundries, and a more educated black people. Their sacrifices and courage during times of stress provided the foundation for the descendants of slaves to build a life position that was more stable than one built on only financial injections to the family. From this foundation, the family was fortified spiritually, civically and educationally. For from the experiences of her home business, the washwoman learned the know-how that ushered the Negro people into forming business enterprises to serve their communities. From the wisdom she gained from observing the American society, she supported building and maintaining organizations through which many emerged into the mainstream. From her zeal to overcome societies’ inequities, she embraced education as the optimal delivery system to a better life for succeeding generations. To that end, she funded causes that fostered education and societies that fought for her community. But equally important, she upheld the primacy of male leadership within the family as a cornerstone of a strong family unit. Upon that rock, the Black family met and fought the war for civil rights in the ’60s.

To read more click the link. Call to Remember