Determine the value of this: 15P6 = Determine the value of this: 15C6 = Determine the value of this: (17/11) =
The correct answer and explanation is:
Let’s evaluate the following expressions:
1. 15P615P6 — Permutation
Definition:
Permutations represent the number of ways to arrange r items out of n items in order.
Formula: nPr=n!(n−r)!nPr = \frac{n!}{(n – r)!}
Calculation: 15P6=15!(15−6)!=15!9!15P6 = \frac{15!}{(15 – 6)!} = \frac{15!}{9!}
Now calculate: 15P6=15×14×13×12×11×10=3,603,60015P6 = 15 \times 14 \times 13 \times 12 \times 11 \times 10 = 3,603,600
2. 15C615C6 — Combination
Definition:
Combinations represent the number of ways to choose r items from n items without regard to order.
Formula: nCr=n!r!(n−r)!nCr = \frac{n!}{r!(n – r)!}
Calculation: 15C6=15!6!⋅(15−6)!=15!6!⋅9!15C6 = \frac{15!}{6! \cdot (15 – 6)!} = \frac{15!}{6! \cdot 9!}
Compute: 15C6=15×14×13×12×11×106×5×4×3×2×1=3,603,600720=5,00515C6 = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = \frac{3,603,600}{720} = 5,005
3. 1711\frac{17}{11} — Fraction to Decimal
Calculation: 1711=1.545454545…=1.54‾(repeating decimal)\frac{17}{11} = 1.545454545… = 1.\overline{54} \quad (\text{repeating decimal})
Explanation (300 Words Approx.)
In mathematics, permutations and combinations are essential concepts in probability and counting problems. Permutations, denoted as nPrnPr, are used when the order of the selected items matters. For example, arranging 6 people in a line from a group of 15 means each arrangement is different depending on who comes first, second, and so on. The formula for permutations is n!(n−r)!\frac{n!}{(n-r)!}. For 15P615P6, we use n=15n = 15, r=6r = 6, and find 15!9!\frac{15!}{9!}, which simplifies to multiplying 15 down to 10, yielding 3,603,600 ways to arrange 6 items from 15.
Combinations, denoted as nCrnCr, are used when order does not matter — such as choosing 6 lottery numbers from 15, where the order of drawing doesn’t change the outcome. The formula is n!r!(n−r)!\frac{n!}{r!(n-r)!}, and for 15C615C6, we get 15!6!⋅9!=5,005\frac{15!}{6! \cdot 9!} = 5,005. This tells us there are 5,005 unique ways to choose 6 items from 15 without regard to order.
Finally, 1711\frac{17}{11} is a basic fraction. When converted to a decimal, it gives a repeating value: 1.545454…, which is written as 1.54‾1.\overline{54}. Understanding how to convert fractions to decimals is crucial in many areas of math, including statistics and measurement.
These three topics — permutations, combinations, and fractional values — are foundational for solving real-world problems involving probability, arrangements, and number sense.