$\frac{9}{6 m}_{6 m}^{9 c m} 3 m$
Answer (2)
The simplified expression of the given mathematical equation is 378. This was achieved by calculating the binomial coefficient, substituting it into the expression, and simplifying. The final answer is 378 .
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Calculate the binomial coefficient: ( 6 9 ) = 84 .
Substitute the binomial coefficient into the expression: 6 m 9 ⋅ 84 ⋅ 3 m .
Simplify the expression by cancelling m : 6 9 ⋅ 84 ⋅ 3 .
Calculate the final result: 378 . The simplified expression is 378 .
Explanation
Understanding the Expression We are given the expression 6 m 9 ( 6 m 9 c m ) 3 m . Our goal is to simplify this expression.
Assumptions and Simplifications First, let's consider the binomial coefficient ( 6 m 9 c m ) . It is likely that 'cm' is just a typo and the expression is actually ( 6 9 ) . Also, we assume that m is a dimensionless variable.
Calculating the Binomial Coefficient We calculate the binomial coefficient ( 6 9 ) using the formula ( k n ) = k ! ( n − k )! n ! . Thus, we have ( 6 9 ) = 6 ! 3 ! 9 ! = 6 ! × 3 × 2 × 1 9 × 8 × 7 × 6 ! = 3 × 2 × 1 9 × 8 × 7 = 6 9 × 8 × 7 = 3 × 4 × 7 = 84.
Substituting the Result Now we substitute the value of the binomial coefficient back into the original expression: 6 m 9 ( 6 9 ) 3 m = 6 m 9 ⋅ 84 ⋅ 3 m = 6 m 9 ⋅ 84 ⋅ 3 m .
Simplifying the Expression We simplify the expression by cancelling out the common factor m and then simplifying the numbers: 6 m 9 ⋅ 84 ⋅ 3 m = 6 9 ⋅ 84 ⋅ 3 = 2 9 ⋅ 84 = 9 ⋅ 42 = 378.
Final Answer Therefore, the simplified expression is 378.
Examples
Binomial coefficients are used in probability calculations, such as determining the likelihood of winning a lottery or drawing specific cards from a deck. Simplifying expressions involving binomial coefficients can help in calculating probabilities more efficiently. For example, if you want to know the probability of drawing exactly 3 aces from a deck of 52 cards when drawing 5 cards, you would use binomial coefficients to calculate the number of ways to choose 3 aces from 4 and 2 other cards from the remaining 48. Simplifying such expressions makes the calculation easier.
