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Evaluate \newline9C6_{9}C_{6} and \newline11P4_{11}P_{4}.\newline\begin{array}{l} _{9}C_{6}=\Box,\ _{11}P_{4}=\Box \end{array}

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Add and subtract three or more integersright chevron icon

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Q. Evaluate \newline9C6_{9}C_{6} and \newline11P4_{11}P_{4}.\newline\begin{array}{l} _{9}C_{6}=\Box,\ _{11}P_{4}=\Box \end{array}
  1. Evaluate Combination: Evaluate (9)C(6)_(9)C_(6), which represents the number of combinations of 99 items taken 66 at a time.\newlineThe formula for combinations is:\newline(n)C(r)=n!r!(n−r)!_(n)C_(r) = \frac{n!}{r!(n - r)!}\newlinewhere n!n! denotes the factorial of nn.\newlineLet's calculate (9)C(6)_(9)C_(6) using the formula.\newline(9)C(6)=9!6!(9−6)!_(9)C_(6) = \frac{9!}{6!(9 - 6)!}
  2. Simplify Combination: Simplify the expression for 9C6_{9}C_{6}.9C6=9!6!3!_{9}C_{6} = \frac{9!}{6!3!}9!=9×8×7×6!9! = 9 \times 8 \times 7 \times 6!3!=3×2×13! = 3 \times 2 \times 1Now, we can cancel out the common 6!6! term in the numerator and denominator.9C6=9×8×73×2×1_{9}C_{6} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1}
  3. Calculate Combination: Perform the calculation for (9)C(6)_(9)C_(6).
    (9)C(6)=9×8×73×2×1_(9)C_(6) = \frac{9 \times 8 \times 7}{3 \times 2 \times 1}
    (9)C(6)=5046_(9)C_(6) = \frac{504}{6}
    (9)C(6)=84_(9)C_(6) = 84
  4. Evaluate Permutation: Evaluate 11P4_{11}P_{4}, which represents the number of permutations of 1111 items taken 44 at a time.\newlineThe formula for permutations is:\newlinenPr=n!(n−r)!_{n}P_{r} = \frac{n!}{(n - r)!}\newlineLet's calculate 11P4_{11}P_{4} using the formula.\newline11P4=11!(11−4)!_{11}P_{4} = \frac{11!}{(11 - 4)!}
  5. Simplify Permutation: Simplify the expression for 11P4_{11}P_{4}.11P4=11!7!_{11}P_{4} = \frac{11!}{7!}11!=11×10×9×8×7!11! = 11 \times 10 \times 9 \times 8 \times 7!Now, we can cancel out the common 7!7! term in the numerator and denominator.11P4=11×10×9×8_{11}P_{4} = 11 \times 10 \times 9 \times 8
  6. Calculate Permutation: Perform the calculation for (11P4)(_{11}P_{4}).\newline(11P4)=11×10×9×8(_{11}P_{4}) = 11 \times 10 \times 9 \times 8\newline(11P4)=7920(_{11}P_{4}) = 7920

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