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¿De cuántas formas te puedes vestir?

Imagina que te levantas un día por la mañana para ir a trabajar, abres tu guardarropa y observas que dispones de una camisa azul, una blanca, una roja y una verde, también ves tus pantalones negros de vestir, tu pantalón caqui y tus jeans favoritos. Cuando bajas la vista observas que están tus zapatos negros y los de color café. Teniendo tantas opciones, ¿de cuántas maneras podrías vestirte para ir a trabajar?

Para responder a esta pregunta podrías usar varias estrategias, veámos algunas a continuación

Diagrama de árbol

Podrías comenzar dibujando un "arbolito", para cada opción dibujaremos una ramita y terminaremos contando cuántas ramas tiene el arbolito al final.

Arbolito
Arbolito

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En este caso observamos que el arbolito tiene 24 ramas en total, 6 azules, 6 negras, 6 rojas y 6 verdes.

Enlistar las opciones

Para organizar mejor nuestro listado hagamos una tabla, y para simplificar la escritura asignemos una letra a cada opción así: Camisa azul (A), camisa blanca (B), camisa roja (R), camisa verde (V), pantalón negro (P), pantalón caqui (M), jeans (J), zapatos negros (N), zapatos cafés (C).

APN BPN RPN VPN
APC BPC RPC VPC
AMN BMN RMN VMN
AMC BMC RMC VMC
AJN BJN RJN VJN
AJC BJC RJC VJC

Contamos las opciones y vemos que también son 24.

Usar el Principio General de la Multiplicación

Comencemos colocando tres lineas horizontales, donde cada una represente alguno de los objetos que debemos elegir.

________   ________   ________

Así la primera línea representa las camisas, la segunda los pantalones y la tercera los zapatos. Ahora, sobre cada una de ellas escribiremos las opciones que tenemos para escoger, de la siguiente manera

  4        3       2  

Ahora, como para cada camisa tenemos tres pantalones para ponernos deberíamos multiplicar las opciones para saber cuántas opciones combinadas hay, pero además para cada pantalón podemos usar dos pares distintos de zapatos, también debemos multiplicar, de forma que el resultado es el producto de todas las opciones disponibles.

\(4\cdot3\cdot2=24\)

Para reflexionar

Pensemos

Seguro se te ocurren más formas de resolver esta situación pero concentrémonos en las tres presentadas. Reflexiona sobre las siguientes preguntas.

¿Cuál es la forma correcta de resolver el problema?
¿Consideras que alguna de las formas de resolver es más sencilla que otra?
¿Qué pasaría con el arbolito con un guardarropa que tiene 100 camisas, 20 pantalones y 12 pares de zapatos?
¿Qué pasaría con la tabla con un guardarropa que tiene 100 camisas, 20 pantalones y 12 pares de zapatos?
¿Cómo lo resolverías usando el Principio General de la Multiplicación?

Lectura facilitada

Observa que todas las formas de realizar la actividad son correctas, sin embargo, algunas aunque representen ideas más sencillas se vuelven más complicadas de ejecutar conforme van aumentando el número de posibilidades, observa el tamaño del arbolito, con 100 camisas definitivamente no cabría en una hoja de libreta. Una tabla aunque pueda ser más organizada y compacta se vuelve difícil porque debes ser muy cuidadoso al armar cada una de las opciones, porque no debes repetir ni olvidar ninguna.

Entonces, el Principio General de la Multiplicación se vuelve una herramienta muy potente y útil. Lo enunciaremos y aprovecharemos más adelante.

Definición

Principio General de la Multiplicación

Supóngase que una experiencia \(A\) puede arrojar \(m\) resultados, y por cada uno de éstos una experiencia \(B\) puede arrojar \(n \) resultados. Entonces la realización conjunta de \(A\) y \(B\) pude arrojar \(m\times n\) resultados.

Lectura facilitada

Aquí tienes más material para consultar, de la Olimpiada Mexicana de Matemáticas en Baja California: Principios de Combinatoria

Apoyo visual

Puedes ver este video sobre el Principio General de la Multiplicación

Autor: Matemáticas Profe Alex

Inténtalo

¿De cuántas formas distintas puede vestirse una persona con 4 camisas y 3 pantalones como los mostrados?

Lectura facilitada

Dibuja en una hoja blanca 4 camisas y 3 pantalones como los de las imágenes, colorea tus dibujos, cada uno de un color distinto y recórtalos para armar todas las vestimentas posibles.

Creado con eXeLearning (Ventana nueva)