Número Binomial
é a representação do número binomial de numerador n e denominador k, ou então do número binomial n sobre k.
Sendo n ≥ k, com n e k pertencentes ao conjunto do números naturais, o número binomial
é dado por:
![](MEx.ashx?XGxlZnQoXGJlZ2lue2FycmF5fXtjfSBuXFwga1xlbmR7YXJyYXl9XHJpZ2h0KVxxdWFkPVxxdWFkIEN7X3sgbl8sXHF1YWQga319XHFxdWFkXFJpZ2h0YXJyb3dccXF1YWRcbGVmdChcYmVnaW57YXJyYXl9e2N9IG5cXCBrXGVuZHthcnJheX1ccmlnaHQpXHF1YWQ9XHF1YWRcZnJhY3tuIX17ayEoblxxdWFkLVxxdWFkIGspIX0=)
Note que os números binomiais, ou coeficientes binomiais, são obtidos através da combinações simples de n elementos distintos, agrupados k a k, com k ≤ n.
Realizando a simplificação de fatoriais chegamos a esta outra expressão:
![](MEx.ashx?XGxlZnQoXGJlZ2lue2FycmF5fXtjfSBuXFwga1xlbmR7YXJyYXl9XHJpZ2h0KVxxdWFkPVxxdWFkXGZyYWN7biF9e2shKG5ccXVhZC1ccXVhZCBrKSF9XHF1YWRcUmlnaHRhcnJvd1xxdWFkXGxlZnQoXGJlZ2lue2FycmF5fXtjfW5cXFxcXFxrXGVuZHthcnJheX1ccmlnaHQpXHF1YWQ9XHF1YWRcZnJhY3tuKG5ccXVhZC1ccXVhZDEpKG5ccXVhZC1ccXVhZDIpXHF1YWRcY2RvdHNccXVhZChuXHF1YWQtXHF1YWQga1xxdWFkK1xxdWFkMSkoblxxdWFkLVxxdWFkIGspIX17ayEoblxxdWFkLVxxdWFkIGspIX1ccXVhZFxSaWdodGFycm93XFxccXVhZFxcXHF1YWRcXFxxdWFkXFxcUmlnaHRhcnJvd1xxdWFkXGxlZnQoXGJlZ2lue2FycmF5fXtjfW5cXFxcXFxrXGVuZHthcnJheX1ccmlnaHQpXHF1YWQ9XHF1YWRcZnJhY3tuKG5ccXVhZC1ccXVhZDEpKG5ccXVhZC1ccXVhZDIpXHF1YWRcY2RvdHNccXVhZChuXHF1YWQtXHF1YWQga1xxdWFkK1xxdWFkMSl9e2shfQ==)
Alguns números binomiais podem ser obtidos sem precisarmos realizar o cálculo da combinação simples, por exemplo:
![](images/trianglered.gif)
, pois: ![](MEx.ashx?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)
![](images/trianglered.gif)
, pois: ![](MEx.ashx?XGxlZnQoXGJlZ2lue2FycmF5fXtjfSBuXFxcXFxcMVxlbmR7YXJyYXl9XHJpZ2h0KVxxdWFkPVxxdWFkXGZyYWN7biF9ezEhKG5ccXVhZC1ccXVhZDEpIX1ccXVhZFxSaWdodGFycm93XHF1YWRcbGVmdChcYmVnaW57YXJyYXl9e2N9IG5cXFxcXFwxXGVuZHthcnJheX1ccmlnaHQpXHF1YWQ9XHF1YWRcZnJhY3tuXHF1YWRcY2RvdFxxdWFkKG5ccXVhZC1ccXVhZDEpIX17MShuXHF1YWQtXHF1YWQxKSF9XHF1YWRcUmlnaHRhcnJvd1xxdWFkXGxlZnQoXGJlZ2lue2FycmF5fXtjfSBuXFxcXFxcMVxlbmR7YXJyYXl9XHJpZ2h0KVxxdWFkPVxxdWFkIG4=)
![](images/trianglered.gif)
, pois: ![](MEx.ashx?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)
Igualdade de Números Binomiais
Dois números binomiais
e
, com
, são iguais se:
k = j, ou -
k + j = n-
Segundo a propriedade dos termos complementares, se fizermos j = n - k, temos ainda que:
![](MEx.ashx?XGxlZnQoXGJlZ2lue2FycmF5fXtjfW5cXFxcXFxrXGVuZHthcnJheX1ccmlnaHQpXHF1YWQ9XHF1YWRcbGVmdChcYmVnaW57YXJyYXl9e2N9blxcXFxcXGpcZW5ke2FycmF5fVxyaWdodClccXVhZFxSaWdodGFycm93XHF1YWRcbGVmdChcYmVnaW57YXJyYXl9e2N9blxcXFxcXGtcZW5ke2FycmF5fVxyaWdodClccXVhZD1ccXVhZFxsZWZ0KFxiZWdpbnthcnJheX17Y31uXFxcXFxcblxxdWFkLVxxdWFkIGtcZW5ke2FycmF5fVxyaWdodCk=)
Em outras palavras, para que dois números binomiais
e
sejam iguais, k e j devem ser iguais ou complementares.
Exemplos de Igualdade entre Números Binomiais
![](images/trianglered.gif)
![](MEx.ashx?XGxlZnQoXGJlZ2lue2FycmF5fXtjfTdcXFxcXFw1XGVuZHthcnJheX1ccmlnaHQpXHF1YWQ9XHF1YWRcbGVmdChcYmVnaW57YXJyYXl9e2N9N1xcXFxcXDVcZW5ke2FycmF5fVxyaWdodCk=)
![](images/trianglered.gif)
![](MEx.ashx?XGxlZnQoXGJlZ2lue2FycmF5fXtjfTlcXFxcXFwzXGVuZHthcnJheX1ccmlnaHQpXHF1YWQ9XHF1YWRcbGVmdChcYmVnaW57YXJyYXl9e2N9OVxcXFxcXDZcZW5ke2FycmF5fVxyaWdodCk=)
![](images/trianglered.gif)
![](MEx.ashx?XGxlZnQoXGJlZ2lue2FycmF5fXtjfThcXFxcXFwyXGVuZHthcnJheX1ccmlnaHQpXHF1YWQ9XHF1YWRcbGVmdChcYmVnaW57YXJyYXl9e2N9OFxcXFxcXDZcZW5ke2FycmF5fVxyaWdodCk=)
![](images/trianglered.gif)
![](MEx.ashx?XGxlZnQoXGJlZ2lue2FycmF5fXtjfTRcXFxcXFw0XGVuZHthcnJheX1ccmlnaHQpXHF1YWQ9XHF1YWRcbGVmdChcYmVnaW57YXJyYXl9e2N9NFxcXFxcXDBcZW5ke2FycmF5fVxyaWdodCk=)
Relação de Stifel
A Relação de Stifel ou Regra de Pascal, nos permite realizar a seguinte igualdade:
![](MEx.ashx?XGxlZnQoXGJlZ2lue2FycmF5fXtjfW5ccXVhZC1ccXVhZDFcXFxcXFxrXHF1YWQtXHF1YWQxXGVuZHthcnJheX1ccmlnaHQpXHF1YWQrXHF1YWRcbGVmdChcYmVnaW57YXJyYXl9e2N9blxxdWFkLVxxdWFkMVxcXFxcXGtcZW5ke2FycmF5fVxyaWdodClccXVhZD1ccXVhZFxsZWZ0KFxiZWdpbnthcnJheX17Y31uXFxcXFxcXFxrXGVuZHthcnJheX1ccmlnaHQp)
Esta regra é válida para: ![](MEx.ashx?bixccXVhZCBrXHF1YWRcaW5ccXVhZFxtYXRoYmJ7Tn1eKixccXVhZCBuXHF1YWQ+XHF1YWQgaw==)
Exemplos Utilizando a Relação de Stifel
![](images/trianglered.gif)
![](MEx.ashx?XGxlZnQoXGJlZ2lue2FycmF5fXtjfTEyXFxcXFxcN1xlbmR7YXJyYXl9XHJpZ2h0KVxxdWFkK1xxdWFkXGxlZnQoXGJlZ2lue2FycmF5fXtjfTEyXFxcXFxcOFxlbmR7YXJyYXl9XHJpZ2h0KVxxdWFkPVxxdWFkXGxlZnQoXGJlZ2lue2FycmF5fXtjfTEzXFxcXFxcOFxlbmR7YXJyYXl9XHJpZ2h0KVxxdWFkXFJpZ2h0YXJyb3dccXVhZDc5MlxxdWFkK1xxdWFkNDk1XHF1YWQ9XHF1YWQxMjg3)
![](images/trianglered.gif)
![](MEx.ashx?XGxlZnQoXGJlZ2lue2FycmF5fXtjfTlcXFxcXFw0XGVuZHthcnJheX1ccmlnaHQpXHF1YWQrXHF1YWRcbGVmdChcYmVnaW57YXJyYXl9e2N9OVxcXFxcXDVcZW5ke2FycmF5fVxyaWdodClccXVhZD1ccXVhZFxsZWZ0KFxiZWdpbnthcnJheX17Y30xMFxcXFxcXDVcZW5ke2FycmF5fVxyaWdodClccXVhZFxSaWdodGFycm93XHF1YWQxMjZccXVhZCtccXVhZDEyNlxxdWFkPVxxdWFkMjUy)
![](images/trianglered.gif)
![](MEx.ashx?XGxlZnQoXGJlZ2lue2FycmF5fXtjfTZcXFxcXFwyXGVuZHthcnJheX1ccmlnaHQpXHF1YWQrXHF1YWRcbGVmdChcYmVnaW57YXJyYXl9e2N9NlxcXFxcXDNcZW5ke2FycmF5fVxyaWdodClccXVhZD1ccXVhZFxsZWZ0KFxiZWdpbnthcnJheX17Y303XFxcXFxcM1xlbmR7YXJyYXl9XHJpZ2h0KVxxdWFkXFJpZ2h0YXJyb3dccXVhZDE1XHF1YWQrXHF1YWQyMFxxdWFkPVxxdWFkMzU=)
![](images/h700.gif)