A certain lottery has 27 numbers. In how many different ways can 6 of the numbers be selected? (Assume that order of selection is not important) There are \( \square \) different ways the numbers can be selected (Simplify your answer.)
Solución de inteligencia artificial de Upstudy
Responder
Solución

¡Inicia sesión para desbloquear respuestas gratis!
Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

Respondido por UpStudy AI y revisado por un tutor profesional
Beyond the Answer
To find the number of different ways to select 6 numbers from a set of 27, we use the combination formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] Here, \( n = 27 \) and \( r = 6 \). Plugging in the values, we get: \[ \binom{27}{6} = \frac{27!}{6!(27-6)!} = \frac{27!}{6! \times 21!} \] Now, to simplify: \[ \binom{27}{6} = \frac{27 \times 26 \times 25 \times 24 \times 23 \times 22}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \] Calculating the numerator: \[ 27 \times 26 = 702 \] \[ 702 \times 25 = 17550 \] \[ 17550 \times 24 = 421200 \] \[ 421200 \times 23 = 9687600 \] \[ 9687600 \times 22 = 213210600 \] Now calculating the denominator: \[ 6 \times 5 = 30 \] \[ 30 \times 4 = 120 \] \[ 120 \times 3 = 360 \] \[ 360 \times 2 = 720 \] \[ 720 \times 1 = 720 \] Now dividing the results: \[ \frac{213210600}{720} = 295263.75 \] It seems there was an arithmetic mistake. Let’s carefully recalculate: The correct step to divide is: \[ 213210600 \div 720 = 295263 \] Hence, the total number of different ways to select 6 numbers from 27 is: \[ \binom{27}{6} = 296010 \] So, the answer is: \[ \text{There are } 296010 \text{ different ways the numbers can be selected.} \]
