Worksheet on Combinations and Permutations

Worksheet on Combinations and Permutations Permutations and combinations are two concepts that related to ideas in probability.   These two topics are very similar and are easy to get confused. In both cases we start with a set containing a a total of n elements.   Then we count r of these elements. The way in which we count these elements determines if we are working with a combination or with a permutation. Ordering and Arrangement The key things to remember when distinguishing between combinations and permutations has to do with order and arrangements.   Permutations deal with situations when the order that we choose the objects is important.   We can also think of this as being equivalent to the idea of arranging objects In combinations we are not concerned with what order we selected our objects. We only need this concept, and the formulas for combinations and permutations to solve problems dealing with this topic. Practice Problems To get good at something, it takes some practice.   Here are some practice problems with solutions to help you to straighten out the ideas of permutations and combinations. A version with answers is here.   After starting with just basic calculations, you can use what you know to determine if a combination or permutation is being referred to. Use the formula for permutations to calculate P( 5, 2 ).Use the formula for combinations to calculate  C( 5, 2 ).Use the formula for permutations to calculate  P( 6, 6 ).Use the formula for combinations to calculate  C( 6, 6 ).Use the formula for permutations to calculate  P( 100, 97 ).Use the formula for combinations to calculate  C( 100, 97 ).It’s election time at a high school that has a total of 50 students in the junior class. How many ways can a class president, class vice president, class treasurer,and class secretary be chosen if each student may only hold one office?The same class of 50 students wants to form a prom committee. How many ways can a four person prom committee be selected from the junior class?If we want to form a group of five students and we have 20 to choose from, how many ways is this possible?How many ways can we arrange four letters from the word â€Å"computer† if repetitions are not allowed, and different orders of the same letter s count as different arrangements?How many ways can we arrange four letters from the word â€Å"computer† if repetitions are not allowed, and different orders of the same letters count as the same arrangement? How many different four digit numbers are possible if we can choose any digits from 0 to 9 and all of the digits must be different?If we are given a box containing seven books, how many ways can we arrange three of them on a shelf?If we are given a box containing seven books, how many ways can we choose collections of three of them from the box?

A Brief Biography of Gregor Mendel, Father of Genetics

A Brief Biography of Gregor Mendel, Father of Genetics Gregor Mendel, known as the Father of Genetics, is most well-known for his work with breeding and cultivating pea plants, using them to gather data about dominant and recessive genes. Dates: Born July 20, 1822 - Died January 6, 1884 Early Life and Education Johann Mendel was born in 1822 in the Austrian Empire to Anton Mendel and Rosine Schwirtlich. He was the only boy in the family and worked on the family farm with his older sister Veronica and his younger sister Theresia. Mendel took an interest in gardening and beekeeping as he grew up. As a young boy, Mendel attended school in Opava. He went on to the University of Olomouc after graduating, where he studied many disciplines, including physics and philosophy. He attended the University from 1840 to 1843 and was forced to take a year off due to illness. In 1843, he followed his calling into the priesthood and entered the Augustinian Abbey of St. Thomas in Brno. Personal Life Upon entering the Abbey, Johann took the first name Gregor as a symbol of his religious life. He was sent to study at the University of Vienna in 1851 and returned to the abbey as a teacher of physics. Gregor also cared for the garden and had a set of bees on the abbey grounds. In 1867, Mendel was made an abbot of the abbey. Genetics Gregor Mendel is best known for his work with his pea plants in the abbey gardens. He spent about seven years planting, breeding and cultivating pea plants in an experimental part of the abbey garden that was started by the previous abbot. Through meticulous record keeping, Mendels experiments with pea plants became the basis for modern genetics. Mendel chose pea plants as his experimental plant for many reasons. First of all, pea plants take very little outside care and grow quickly. They also have both male and female reproductive parts, so they can either cross-pollinate or self-pollinate. Perhaps most importantly, pea plants seem to show one of only two variations of many characteristics. This made the data much more clear-cut and easier to work with. Mendels first experiments focused on one trait at a time, and on gathering data on the variations present for several generations. These were called monohybrid experiments. He studied a total of seven characteristics. His findings showed that there were some variations that were more likely to show up over the other variations. When he bred purebred peas of differing variations, he found that in the next generation of pea plants one of the variations disappeared. When that generation was left to self-pollinate, the next generation showed a 3 to 1 ratio of the variations. He called the one that seemed to be missing from the first filial generation recessive and the other dominant, since it seemed to hide the other characteristic. These observations led Mendel to the law of segregation. He proposed that each characteristic was controlled by two alleles, one from the mother and one from the father plant. The offspring would show the variation it is coded for by the dominance of the alleles. If there is no dominant allele present, then the offspring shows the characteristic of the recessive allele. These alleles are passed down randomly during fertilization. Link to Evolution Mendels work wasnt truly appreciated until the 1900s, long after his death. Mendel had unknowingly provided the Theory of Evolution with a mechanism for the passing down of traits during natural selection. As a man of strong religious conviction, Mendel did not believe in evolution during his life. However, his work has been added together with that of Charles Darwins to make up the modern synthesis of the Theory of Evolution. Much of Mendels early work in genetics has paved the way for modern scientists working in the field of microevolution.